Focusing strictly on the question of the Biblical basis for believing that breaking the civil laws of the State is sinful, here are some of the Bible passages implying or stating that we should obey the civil authorities (which, in the case of the ancient Jewish State, were the same as the religious authorities).

These are quoted in their Biblical order.

First from the Gospels:

When Jesus had come down from the mountain, great crowds followed him; and there was a leper who came to him and knelt before him, saying, "Lord, if you choose, you can make me clean."

He stretched out his hand and touched him, saying, "I do choose. Be made clean!" Immediately his leprosy was cleansed. Then Jesus said to him, "See that you say nothing to anyone; but go, show yourself to the priest, and offer the gift that Moses commanded, as a testimony to them." (Matthew 8:1-4; see also the parallel passages in Mark 1:40-45; Luke 5:12-16)

Here Jesus instructs a man he has healed to go to the priest and offer the gift commanded in the Law of Moses, "as a testimony to them." This begins to establish a theme of obeying the authorities as a way of testifying to the power and sovereignty of God.

The next passage provides a different angle on this theme:

When they reached Capernaum, the collectors of the temple tax came to Peter and said, "Does your teacher not pay the temple tax?"

He said, "Yes, he does."

And when he came home, Jesus spoke of it first, asking, "What do you think, Simon? From whom do kings of the earth take toll or tribute? From their children or from others?"

Peter said, "From others."

Jesus said to him, "Then the children are free. However, so that we do not give offense to them, go to the sea and cast a hook; take the first fish that comes up; and when you open its mouth, you will find a coin; take that and give it to them for you and me." (Matthew 17:24-27)

Here Jesus argues that "the children," meaning citizens of the nation, are free from the taxes imposed by the authorities—in this case, the religious authorities. Yet he tells Peter to pay the temple tax anyway, "so that we do not give offense to them." This is commonly read as establishing a theme of obeying civil law so as not to offend against the civil authorities and the sensibilities of the people, who hold the civil law in esteem for keeping civil order and punishing evildoers.

This theme is developed more fully in 1 Peter 2:11-17, quoted in its Biblical order below.

Then the Pharisees went and plotted to entrap him in what he said. So they sent their disciples to him, along with the Herodians, saying, "Teacher, we know that you are sincere, and teach the way of God in accordance with truth, and show deference to no one; for you do not regard people with partiality. Tell us, then, what you think. Is it lawful to pay taxes to the emperor, or not?"

But Jesus, aware of their malice, said, "Why are you putting me to the test, you hypocrites? Show me the coin used for the tax." And they brought him a denarius. Then he said to them, "Whose head is this, and whose title?"

They answered, "The emperor's."

Then he said to them, "Give therefore to the emperor the things that are the emperor's, and to God the things that are God's." When they heard this, they were amazed; and they left him and went away. (Matthew 22:15-22; see also the parallel passages in Luke 20:20-26; Mark 12:13-17)

Though the message here is nuanced, it is commonly interpreted to mean that as long as we are under a civil authority, we should pay the taxes required by the civil authority, and in broader terms, obey the civil authorities in matters in which it has jurisdiction, while also obeying the law of God where it has jurisdiction (which would presumably be in all significant matters of conduct).

Then Jesus said to the crowds and to his disciples, "The scribes and the Pharisees sit on Moses' seat; therefore, do whatever they teach you and follow it." (Matthew 23:1-3)

Here Jesus tells his hearers to obey the currently sitting religious authorities, even though in other places he commonly refutes their interpretation of the law and states that they have nullified the law by their human edicts.

And from the Epistles:

Let every person be subject to the governing authorities; for there is no authority except from God, and those authorities that exist have been instituted by God. Therefore whoever resists authority resists what God has appointed, and those who resist will incur judgment. For rulers are not a terror to good conduct, but to bad. Do you wish to have no fear of the authority? Then do what is good, and you will receive its approval; for it is God’s servant for your good. But if you do what is wrong, you should be afraid, for the authority does not bear the sword in vain! It is the servant of God to execute wrath on the wrongdoer. Therefore one must be subject, not only because of wrath but also because of conscience. (Romans 13:1-5)

This one the clearest and most extensive Biblical injunctions to obey the civil authorities.

Its basis is different than that of many of the other passages quoted in this answer. Rather than stating that we should obey the civil authorities so as not to give offense, and so as to be examples of good and law-abiding citizens, thus drawing people to God, this passage states that God has established the civil authorities in their positions, and that obeying them is obeying God's agents. Failing to do so will bring judgment down upon the heads of the lawbreakers.

This could be interpreted as incurring civil judgment and penalties, or as incurring divine judgment—which would imply that breaking the civil law is sinful, and will be subject to God's judgment.

A brief but on-point statement:

Remind them to be subject to rulers and authorities, to be obedient, to be ready for every good work. (Titus 3:1)

Another clear and extensive Biblical injunction to accept and obey civil authorities:

Beloved, I urge you as aliens and exiles to abstain from the desires of the flesh that wage war against the soul. Conduct yourselves honorably among the Gentiles, so that, though they malign you as evildoers, they may see your honorable deeds and glorify God when he comes to judge.

For the Lord's sake accept the authority of every human institution, whether of the emperor as supreme, or of governors, as sent by him to punish those who do wrong and to praise those who do right. For it is God's will that by doing right you should silence the ignorance of the foolish. As servants of God, live as free people, yet do not use your freedom as a pretext for evil. Honor everyone. Love the family of believers. Fear God. Honor the emperor. (1 Peter 2:11-17)

Notice that although this passage does echo the theme of Romans 13:1-5 in saying that the civil authorities are God's agents, "sent by him to punish those who do wrong and to praise those who do right," it focuses more on obeying civil authorities to maintain a status and reputation as honorable and law-abiding people among the Gentiles, so that they will thereby honor God and be drawn toward God. This draws on themes embodied in the quotes from the Gospels given and commented on above.

None of these passages states explicitly, "If you break the civil law, you are committing a sin against God"—though Romans 13:1-5 does come very close to saying exactly that.

However, these passages do establish a general principle and a commandment that the laws established by the civil authorities should be obeyed. This at least strongly implies that if we disobey civil authorities in matters that do not conflict with divine law, we are, in fact, disobeying divine law because we are disobeying the Biblical injunctions to honor and obey the civil authorities.

It is beyond the scope of the question to discuss exactly when it is legitimate to disobey civil authority. But the general answer is as suggested just above: where civil law clearly and egregiously conflicts with divine law, divine law must take precedence for the believer. By the same token, the believer must be ready and willing to take the consequences of disobeying civil authorities, which may mean social ostracism, fines, prison, or even death.

“Illegal” is not the same thing as “sinful.”

An action is illegal if it contravenes a law established by man, such as the statutes of the Federal, state, or local governments (or whatever juridical order the reader belongs to).

An action is immoral (or objectively sinful) if it contravenes God’s law, which need not coincide in every case with the human law.

Note that, in order for a person actually to commit a sin, he must, evidently, know that what he is doing is wrong (i.e., a contravention of God’s law), and be free to perform that action willingly (e.g., someone who commits a bad action at gunpoint is probably not guilty of a sin).

However, all things being equal, we do have a duty to obey the just laws that are established by our government. That is because, as St. Paul explains, governments receive their authority from God, and when they rule justly, they are actually ruling on behalf of God:

¹ Let every person be subject to the governing authorities. For there is no authority except from God, and those that exist have been instituted by God. ² Therefore whoever resists the authorities resists what God has appointed, and those who resist will incur judgment. ³ For rulers are not a terror to good conduct, but to bad. Would you have no fear of the one who is in authority? Then do what is good, and you will receive his approval, ⁴ for he is God's servant for your good. But if you do wrong, be afraid, for he does not bear the sword in vain. For he is the servant of God, an avenger who carries out God's wrath on the wrongdoer (Romans 13:1-4, ESV).

However, the submission to authority that St. Paul talks about presupposes that the authority is acting justly. If, on the contrary, a government were to mandate doing something objectively sinful (say, for example, a law that makes it illegal to serve citizens belonging to a certain race), then it would actually be the duty of citizens to disobey the civil law; God’s law is always higher than man’s law. Indeed, an unjust law (such as the one in my example) is not really a law at all. This idea is supported by the following passage in Acts:

And the high priest [a legitimate authority in Jerusalem] questioned them [the apostles], ²⁸ saying, “We strictly charged you not to teach in this name, yet here you have filled Jerusalem with your teaching, and you intend to bring this man's blood upon us.” ²⁹ But Peter and the apostles answered, “We must obey God rather than men” (Acts 5:27-29, ESV).

Note that laws are also unjust if they are excessively burdensome, or if they are impossible to fulfill; for example, a law that declares illegal something that is, in fact, good and holy. In such a case, there is not necessarily a strict duty to disobey this kind of law, but there is also no strict obligation to obey it.

In summary, the relationship between “illegal” and “sinful” can be characterized as follows:

• Any action that contravenes God’s law is objectively sinful. Only a small fraction of these actions are regulated by human law at all. In other words, many actions are sinful but not illegal as such.
• A just law should in general be obeyed (keeping in mind that no human law is perfect—there might be cases that the legislator did not forsee). Something justly declared illegal in the law would also be sinful.
• An unjust law need not be obeyed, and if the law actually promotes sinful behavior, it actually must be disobeyed. In this case, that which is technically illegal is not sinful, and might even be dutiful.

In all these cases, we are referring to the objective sinfulness of the actions in question; the person acting must also have knowledge and freedom in order to commit an actual sin.

UPDATE: I will address some of the other concerns brought up by the original poster.

The origin of right and wrong: the so-called “natural law”

The rightness or wrongness of an action does not have its basis in the legality or illegality of it. As I mentioned, legality and especially illegality depend on the explicit mention of the action in question in statutes.

However, the rightness or wrongness of an action is ultimately determined by God, not by the government. In fact, rightness and wrongness exist even in the complete absense of human law. As St. Paul says,

For when Gentiles, who do not have the law, by nature do what the law requires, they are a law to themselves, even though they do not have the law (Romans 2:14).

This is the Biblical basis for the so-called natural law: the idea that human nature itself (which is unchanging in time and common to all human beings) provides us with the information we need to know right from wrong. In this case the “law” that St. Paul is talking about is the Jewish Torah (which provided, among other things, the Ten Commandments to help guild the ancient Israelites’ morality), but the principle is applicable to any kind of law. I do not need the government to tell me that murder is wrong, although I am grateful to the government for making it a crime punishible by law. More importantly, it is not the government that makes murder wrong—God does that by creating us the way He does. Rather, the statute merely renders the natural law explicit.

That is why it is so important for laws to be just: if they do not conform to the natural law—to the morality that God has endowed in our very natures (which is what justice entails)—then they cause confusion and misery. And, as I mentioned, non-conforming (i.e., unjust) laws are not even binding, because in making such laws a state abuses the authority that God has entrusted to it.

Regarding some of the particular sins mentioned by the O.P.

Polygamy

Polygamy is a complex topic in the Bible, but it should be noted that it is only present in the early part of the Old Testament. We see it with some of the Patriarchs (Abraham, Jacob), in the Judges, and with the kings of Israel.

However, even in the Old Testament, there was a growing understanding in the People of Israel that polygamy is not in accordance with God’s plan. It was largely due to his polygamy that Solomon apostatized at the end of his life:

⁴ For when Solomon was old his wives turned away his heart after other gods, and his heart was not wholly true to the Lord his God, as was the heart of David his father. ⁵ For Solomon went after Ashtoreth the goddess of the Sidonians, and after Milcom the abomination of the Ammonites. ⁶ So Solomon did what was evil in the sight of the Lord and did not wholly follow the Lord, as David his father had done (1 Kings 11:4-6, ESV).

Clearly, part of the problem is that he married foreign women, but the situation would never have arisen if he had not married 700 wives (v. 3).

Moreover, we can see from all of the instances of polygamy in the Bible that it is not favorably seen by the Biblical authors. For example polygamy breeds rivalry among the wives or concubines: consider, for instance, the expulsion of Hagar in Gen. 21; the whole episode of Rachel and Leah in Gen. 29, and their in-fighting in Gen. 30; and finally the mistreatment of Hannah (the mother of Samuel) in 1 Samuel 1.

In any case polygamy seems to have fallen out of use in ancient Israel by the time of Christ, and it was certainly the expectation of all the New Testament authors that marriage was between one man and one woman, exclusively.

Hence, the immorality of polygamy preceeds its illegality. Laws that make polygamy illegal simply reflect the natural law in this regard.

Marriage in general

In a similar way, marriage as such is rooted in human nature, and is ultimately determined by God. The state has no authority to define what marriage is: “What therefore God has joined together [i.e., in marriage], let not man separate” (Mark 10:9, ESV).

The government acts unjustly if it attempts to define marriage in a way that is contrary to the nature of marriage (i.e., the pernament, exclusive union between one man and one woman).

Concentration camps

Likewise, if a government were to stipulate that certain classes of persons should be sent to concentration camps, it would be acting unjustly. Citizens would not be bound by such a measure. (The state might, of course, compel such persons to enter a concentration camp by force, but that would be a most grave injustice, obviously.)

Conclusion: the state does not determine morality

In summary, the state absolutely does not determine the morality of our actions. Rather, when it is acting as it should, it merely reflects the morality that already exists, thanks to the human nature that God has given us.

In some cases, the state does have the authority to make decisions about things that are not set in stone (like which side of the street to drive on), and so in general those types of laws should be obeyed.

But laws can only be obeyed if they are in conformity with God’s law, as expressed in our human natures (i.e, the “natural law”).