Does Christianity allow for believing on someone else's behalf?

I will be answering from a Catholic perspective.

Holiness in the Bible

Holiness is univocal in Scripture and refers to sanctity, or the quality of being [more] like God.

1 Peter 1:13-18 (DRB) Wherefore having the loins of your mind girt up, being sober, trust perfectly in the grace which is offered you in the revelation of Jesus Christ, 14 As children of obedience, not fashioned according to the former desires of your ignorance: 15 But according to him that hath called you, who is holy, be you also in all manner of conversation* holy: 16 Because it is written: You shall be holy, for I am holy. 17 And if you invoke as Father him who, without respect of persons, judgeth according to every one's work: converse in fear during the time of your sojourning here.

^{* "Conversation" is archaic English for "conduct; living"}

As you can see, the Apostle explicitly teaches that holiness is 'suitability for heaven' or 'prepared to be judged favorably' by God. The use of the word kadosh (holy) as meaning 'set apart' is contextually evident when it is being used as such. But as you see, God's being 'distinct from others' refers directly to His being blameless.

The Bible teaches that being 'not of this world' is synonymous with living morally, unlike the world.

Jesus teaches this:

Matthew 5:43-48 (DRB) You have heard that it hath been said, Thou shalt love thy neighbour, and hate thy enemy. 44 But I say to you, Love your enemies: do good to them that hate you: and pray for them that persecute and calumniate you: 45 That you may be the children of your Father who is in heaven, who maketh his sun to rise upon the good, and bad, and raineth upon the just and the unjust. 46 For if you love them that love you, what reward shall you have? do not even the publicans this? 47 And if you salute your brethren only, what do you more? do not also the heathens this? 48 Be you therefore perfect, as also your heavenly Father is perfect.

To be like the Father, perfect, is to love your enemy, and to do the right thing. Again, one-to-one relationship between how you live and how you are accounted as holy. And of course it's taken for granted in the Biblical context that perfection is a relative perfection, and the result of the best application of one's good will, not absolute ability to fulfill all righteousness without ever failing.

Not only that, but Jesus makes being a child of God contingent upon such holiness. Some have the audacity to actually claim in some measure that Christ here does not mean 'be perfect,' citing human weakness, and ignoring the grace of God.

There is no modern idea of 'holy because associated with the redemption of Christ.' Traditionally, Christ made one holy by providing the sacraments. Not by literally taking the place of others, which would really be an obstruction of justice, not mercy (providing the means to cure in a man what would otherwise have to be punished). In fact, the Bible explicitly condemns the notion that one set of people are any less prone to the wrath of God than any other. See Romans 2:11-16; 11:23; etc. and also 1 Peter 1:13-18 as cited above, which says, "if you invoke as Father him who, without respect of persons, judgeth according to every one's work: converse in fear during the time of your sojourning here." It isn't a pointless fear or respect, it pertains to our works, which here is given in the form of a warning: 'God judges without respect of who you are based on your works: therefore live this way.'

So there is no question, Biblically speaking, about what holiness means.

Vicarious Merit/Intercession

As for the holiness of one person 'overflowing' to others, this is very Biblical. The passage you cite of a spouse converting the other by their own holiness is proof. Job is another example. Holy Job was the reason mercy was had on his friends, when he prayed for them.

Job 42:7-10 (DRB) And after the Lord had spoken these words to Job, he said to Eliphaz the Themanite: My wrath is kindled against thee, and against thy two friends, because you have not spoken the thing that is right before me, as my servant Job hath. 8 Take unto you therefore seven oxen, and seven rams, and go to my servant Job, and offer for yourselves a holocaust: and my servant Job shall pray for you: his face I will accept, that folly be not imputed to you: for you have not spoken right things before me, as my servant Job hath. 9 So Eliphaz the Themanite, and Baldad the Suhite, and Sophar the Naamathite went, and did as the Lord had spoken to them, and the Lord accepted the face of Job. 10 The Lord also was turned at the [repentance] of Job, when he prayed for his friends. And the Lord gave Job twice as much as he had before.

Through the intercession of holy Job, his friends were spared judgement. Spouses can do the same thing for unbelieving spouses, for example, when they merit the grace otherwise asked for themselves, to be granted to others.

1 Timothy 2:1-2 (DRB) I desire therefore, first of all, that supplications, prayers, intercessions, and thanksgivings be made for all men: 2 For kings, and for all that are in high station: that we may lead a quiet and a peaceable life in all piety and chastity.

Through the prayers of the people of God, graces are merited for others (graces of conversion, healing, exorcisms etc.), which otherwise would not come to them. But the people praying are themselves just instruments of God's grace: they don't earn the response from God, it is the result of a free promise from God Himself that if you pray with faith (which implies asking something "according to the will of God") you will receive (Mt. 21:22; 1 Jn. 5:14)—He is not bound by our asking, only by His own word: "He remains faithful, for He cannot deny Himself" (2 Tim. 2:13).

'Familial Savlation'

Passages which promise if a man and his household believe in Jesus they will be saved do not imply that the grace of forgiveness is granted unwillingly, or without choice, but is meant in the 'if you do x, since y is promised, you will recieve y.' It doesn't mean that all family members will believe, or that all family members will remain believers; some, as Jesus said, will not endure, grow cold, become lukewarm, and fall away because of temptation and the cares of this world (Mt. 24:12-13; Lk. 8:13; 21:34; Rev. 3:16; 2 Pet. 2:21-22; Heb. 12:16-17 etc.). So there is no concept of a whole family being saved just because, for example, their father or mother decided to accept Jesus. On the other hand example plays a large role in your life, and God can place you in a believing family in His inscrutible wisdom, as part of His mercy on you.

If we read that:

Matthew 18:7 (DRB) Woe to the world because of scandals. For it must needs be that scandals come: but nevertheless woe to that man by whom the scandal cometh.

If setting a bad example causes others to sin, although not necessarily, which it does, then setting good example causes people to be good.. although not necessarily.

To someone born into a believing family, more has been given, but much more is at stake if they reject this great grace:

Luke 12:37-48 (DRB) Blessed are those servants, whom the Lord when he cometh, shall find watching. Amen I say to you, that he will gird himself, and make them sit down to meat, and passing will minister unto them. 38 And if he shall come in the second watch, or come in the third watch, and find them so, blessed are those servants. 39 But this know ye, that if the householder did know at what hour the thief would come, he would surely watch, and would not suffer his house to be broken open. 40 Be you then also ready: for at what hour you think not, the Son of man will come. 41 And Peter said to him: Lord, dost thou speak this parable to us, or likewise to all? 42 And the Lord said: Who (thinkest thou) is the faithful and wise steward, whom his lord setteth over his family, to give them their measure of wheat in due season? 43 Blessed is that servant, whom when his lord shall come, he shall find so doing. 44 Verily I say to you, he will set him over all that he possesseth. 45 But if that servant shall say in his heart: My lord is long a coming; and shall begin to strike the menservants and maidservants, and to eat and to drink and be drunk: 46 The lord of that servant will come in the day that he hopeth not, and at the hour that he knoweth not, and shall separate him, and shall appoint him his portion with unbelievers. 47 And that servant who knew the will of his lord, and prepared not himself, and did not according to his will, shall be beaten with many stripes. 48 But he that knew not, and did things worthy of stripes, shall be beaten with few stripes. And unto whomsoever much is given, of him much shall be required: and to whom they have committed much, of him they will demand the more.

The jailer didn't save his family by communicating savedness via matrimonial or familial bond, it says, "and he was baptized at once, he and all his family" (Acts 16:33). So Lydia (ibid. 16:15), so Crispus (18:8). Just as Abraham's children weren't sons of God on account of Abraham's faith alone, but by covenantal circumcision of the flesh, and consequent obedience to God's law, so (the fleshly shadow being done away with) baptism is required by all who want to participate in the New Covenant, "the circumcision made without hands."

Colossians 2:10-12 (DRB) And you are filled in him, who is the head of all principality and power: 11 In whom also you are circumcised with circumcision not made by hand, in despoiling of the body of the flesh, but in the circumcision of Christ: 12 Buried with him in baptism, in whom also you are risen again by the faith of the operation of God, who hath raised him up from the dead.

You ask, does Christianity allow for believing on someone else's behalf? The short answer, from a Protestant view, is no. Simply believing in God (that He exists) or that Jesus is the Son of God does not constitute saving faith. James, leader of the Council of Jerusalem, said this about belief:

You believe that there is one God. Good! Even the demons believe that – and shudder (James 2:19).

There is only one way of salvation, and that is through faith in Jesus Christ:

For it is by grace you have been saved, through faith—and this not from yourselves, it is the gift of God (Ephesians 2:8).

Salvation comes from the Lord (Jonah2:9).

With regard to 1 Corinthians 7:14, the New International Version (NIV) puts it this way:

For the unbelieving husband has been sanctified through his wife, and the unbelieving wife has been sanctified through her believing husband. Otherwise your children would be unclean, but as it is, they are holy.

Being “made holy” is translated here as having “been sanctified” but we must examine the context to understand the meaning. The entire chapter is about marriage and divorce. Verses 12 to 16 specifically address the situation within marriage where only one spouse is a Christian. The NIV Study Bible makes this comment:

The unbelieving partner is influenced by the godly life of the Christian partner; so that family is under the holy influence of the believer and in that sense is sanctified. [The children] at least have the advantage of being under the sanctifying influence of one Christian parent and so may be called holy.

This explanation comes from a Protestant source:

In this context the word sanctified is not referring to salvation or being made holy before God. Instead, it refers to the sanctity of the marriage relationship itself. Paul taught that Christians should not be “unequally yoked” with unbelievers (2 Corinthians 6:14). The fear of some in the church was that, since they were married to unbelievers, they were living in sin—their marriage was “unholy” and their children from that union were illegitimate. Paul allays their fears: believers who are already married to an unbeliever should remain married as long as the unbeliever consents to stay married. They should not seek a divorce; their marriage relationship is sanctified (holy or set apart in God’s eyes) based upon the faith of the believing spouse. Likewise, the children of their marriage are legitimate in the sight of God.

The same article goes on to explain salvation in the context of the Philippian jailer and his household:

The Philippian jailer asks Paul and Silas, “Sirs, what must I do to be saved?” The missionaries respond, “Believe in the Lord Jesus, and you will be saved—you and your household.” Again, this promise is given to a specific individual in a specific context; however, this one contains an additional promise that is clearly universal and spans all time periods and contexts. That promise is not one of household salvation but is entirely consistent with every other verse in the Bible that speaks of salvation. It is the promise that if you believe in the Lord Jesus “you will be saved.” Also, salvation came to the jailer’s household as the result of their hearing the Word of God and individually responding in faith: Paul and Silas “spoke the word of the Lord to him and to all the others in his house” (Acts 16:32). The whole family heard the gospel. They were all saved, just as God had promised, but their salvation was not due to their being a part of the jailer’s household; they were saved because they believed the gospel for themselves. Source: https://www.gotquestions.org/household-salvation.html

Timothy acknowledged that it is the power of God to save and to call us to a holy life (2 Timothy 1:9). Salvation leads to holiness, to a life transformed. Holiness is the result of salvation, but the gift of salvation comes from God alone. No person can bestow salvation on another. We are judged, not on the faith of others, nor on belonging to any religious denomination. The Bible tells us that we are judged according to what we have done while alive (Revelation 20:13).

We are justified by our faith (Romans 5:1) and our faith comes from hearing and believing in the message of Christ (Romans 10:17). That’s why the Philippian jailer’s household was saved – they heard the gospel message and came to saving faith in Christ Jesus.

Everyone who calls on the name of the Lord will be saved (Romans 10:13).

No, the Bible doesn't teach that one person's faith automatically saves others in your family/household. Rather, it teaches that it will be more likely for someone to be saved if they have a family member who can assist them in acquiring and maintaining strong faith.

Parents can show their children the way to life by instructing their children in God's wise commands:

Proverbs 6:20-23 (ESV) - My son, keep your father's commandment, and forsake not your mother's teaching. ²¹  Bind them on your heart always; tie them around your neck. ²²  When you walk, they will lead you; when you lie down, they will watch over you; and when you awake, they will talk with you. ²³ For the commandment is a lamp and the teaching a light, and the reproofs of discipline are the way of life.

A person is not accountable for the decisions of his father or son:

Ezekiel 18:20 (ESV) - The soul who sins shall die. The son shall not suffer for the iniquity of the father, nor the father suffer for the iniquity of the son. The righteousness of the righteous shall be upon himself, and the wickedness of the wicked shall be upon himself.

In the experience at Acts 16:31-34, the jailer wasn't the only one who put faith in Jesus. His whole family did. However, the jailer was the first one of his family to put faith in Jesus, and he brought Paul and Silas to the rest of his household so that they would also learn and acquire faith. If the jailer had not believed in God, the rest of his household would not have heard the life-saving message at that time (or maybe ever), so they rejoiced that the jailer had believed.

Being holy is also different from being saved. The Israelites were holy, because God had chosen them to be his people and to give his commandments (Deut. 7:6), but they were not saved from their enemies unless they were trusting in God and being obedient to his commandments.