Do our fallen, finite nature and our relationship with God set us up for abusive relationships in this life? According to the Catholic Church, yes it does predispose people for suffering and thus for abuse and abusive relationships.

Man sinned, and by doing so forfeited the four-fold harmony by which he was protected from suffering and death. Some suffering is the result of the actions of other people.
(Notice the fourfold harmony: the harmony of friendship between man and God, the harmony within man himself, the harmony between man and woman -or social between human persons, and the harmony between man and the rest of creation).

The Catholic Church does not teach to stay in abusive relationships, furthermore it prohibits fornication, adultery and even chaste unmarried cohabitation but it teaches that you should stay married if you were married -although you can part ways and live separated permanently- unless there is an annulment.

A spouse who occasions grave danger of soul or body to the other or to the children, or otherwise makes the common life unduly difficult, provides the other spouse with a reason to leave, either by a decree of the local ordinary [e.g., bishop] or, if there is danger in delay, even on his or her own authority.

cic 1153 (Catholic Church Code of Canon Law)

Information taken from:

http://www.calledtocommunion.com/2009/08/a-catholic-reflection-on-the-meaning-of-suffering/

http://www.catholic.com/quickquestions/is-divorce-a-sin-when-ones-spouse-is-abusive

http://www.catholic.com/quickquestions/why-is-living-together-wrong-if-were-committed-to-chastity

The Original Poster raises a number of related questions, which have to do with the nature of love and how that applies to human relationships (in particular when they are abusive).

From the outset, it is important to note, as freethinker26 states, that the Catholic Church does not condone or otherwise encourage abusive relationships. In reality, such relationships are a distortion of authentic love, and indeed, authentic supernatural love tends away from such relationships, not toward them. Anyone who finds himself in a situation that places either himself or others (e.g., children) in danger is perfectly within his rights to separate from his spouse, and—if the danger is grave enough—may even be obligated to do so.

The Catechism of the Catholic Church (CCC) states candidly:

Yet there are some situations in which living together becomes practically impossible for a variety of reasons. In such cases the Church permits the physical separation of the couple and their living apart. The spouses do not cease to be husband and wife before God and so are not free to contract a new union. In this difficult situation, the best solution would be, if possible, reconciliation. The Christian community is called to help these persons live out their situation in a Christian manner and in fidelity to their marriage bond which remains indissoluble (No. 1649).

In this same passage, however, the Church teaches that a valid marriage—however abusive one of the spouses has become—remains a marriage, and so the victim should not regard himself as free to marry someone else. (He may, of course, begin the process to investigate the marriage’s nullity: more often than not, the emergence of abusive behavior is a sign of a deeper problem, one likely to have produced circumstances that rendered the exchange of matrimonial consent impossible.)

(Note that reconciliation is recommended when that is possible; in the case of grave and persistent abuse, it is generally not possible.)

The effects of the Fall on married love

To answer the O.P.’s other questions, it is important to understand the effects of the Fall of Adam on mankind. Our First Parents enjoyed sanctifying grace—the gift of friendship with God and adoptive sonship—as well as various praeternatural gifts, including precociousness of the intellect, strength of the will, immortality, and freedom from suffering. The principal effect of the Fall was the rupture of man’s friendship with God—that is, the privation of sanctifying grace. This rupture is by far the most grievous effect of Adam’s sin, and it continues to be what constitutes so-called Original Sin nowadays.

The effect of this rupture with God is that they could no longer love supernaturally: either God or each other. In other words, together with sanctifying grace, they lost the theological virtue of charity.

Moreover, our First Parents also lost the praeternatual gifts: their intellects were darkened, their wills became weaker, and they became particularly prone to sin.

In Baptism, Christians regain sanctifying grace, friendship with God, and the ability to love supernaturally—hence Baptism is the remedy for Original Sin. However, they retain the loss of the praeternatural sin and the tendency to sin.

(Regarding the Fall of Man, see CCC 396-409; regarding sanctifying grace, see CCC 1996-2005.)

The effects of the Fall on marital love

The combination of these losses is at the root of all disorders in marital love. We can see the First Parents began to have marital trouble immediately after their Fall:

Your desire shall be for your husband, and he shall rule over you (Gen. 3, ESV).

Although the Scriptures speak of obedience of a wife to her husband, the authority exercised by the husband was never intended to be tyrannical—the husband is not supposed to rule his wife. Consider, for example

Wives, submit to your own husbands, as to the Lord. For the husband is the head of the wife even as Christ is the head of the church, his body, and is himself its Savior. Now as the church submits to Christ, so also wives should submit in everything to their husbands. Husbands, love your wives, as Christ loved the church and gave himself up for her (Eph. 5:22-25).

and also

For the wife does not have authority over her own body, but the husband does. Likewise the husband does not have authority over his own body, but the wife does (1 Cor. 7:4).

In this last passage St. Paul is speaking about the marital act, and recommends that, generally speaking, spouses should generously give each other their conjugal rights. More broadly, however, it establishes the principle that, although there is a difference in authority between the spouses, it is never an absolute or tyrannical authority, but one founded on Christ’s love for the Church. Indeed, the spouses have, in a certain respects, a reciprocal authority over each other.

Clearly, abuse of any kind is out of the question; it is inimical to marriage as God intended it. Abusive relationships, in fact, arise when marital love is lived out in a disordered way; in no way do they stem from authentic Christian love.

(Regarding marriage under the regime of sin, see CCC 1606-1608.)

Authentic supernatural love, and how it should be lived out in marriage

Fallen man is conceived and born, as I mentioned, deprived of sanctifying grace. That deprivation not only makes it impossible for him to love supernaturally, it produces a tendency in him to sin. In short, fallen man is unable to control his appetities—his tendency towards things that are good in and of themselves—which leads him to neglect greater and more important goods.

Thus, for example, an unfaithful spouse might seek the thrill and pleasure of a new love affair, while sacrificing the far greater good of faithful marriage.

Baptism remedies the privation of sanctifying grace and restores us to friendship with God. Hence, although it does not remove all the effects of that privation (we will retain the troublesome tendency to sin all our lives), the baptized have the strength they need to overcome their sinful tendencies and, more importantly, acquire the ability to love God supernaturally and to love their neighbor for His sake.

It is important to understand, therefore, that authentic Christian love does not consist in a self-emptying for its own sake. It entails self-emptying, certainly, but for the sake of allowing God to act in our souls. In the words of St. Paul,

I have been crucified with Christ. It is no longer I who live, but Christ who lives in me (Gal. 2:20).

That is the goal that we strive for. Indeed, Christian love consists not so much in giving ourselves to others (certainly not “too much of ourselves”) as in giving Christ to others. And we do not have the strength to do that under our own power: it is something that only God can do in us when we have sanctifying grace. (Regarding Christian love, or charity, see CCC 1822-1829.)

Marital love is no different. Christian spouses are called to love one another with supernatural charity. That certainly entail some self-emptying on the part of each spouse, as the spouses will inevitably have differences in tastes, opinion, and habits, and—more importantly—each spouse will inevitably offend the other in some way, hence requiring loving forgiveness from the other spouse.

A marriage lived out in this way does not lead, as such, to an abusive relationship. Both spouses have a responsibility to love the other with the love of Christ (even if, as Paul reminds us, the husband has a special responsibility to love his wife as Christ loves his church; and the wife has a special responsibility to obey her husband as the Church obeys Christ). Neither spouse is free to take advantage of the other selfishly.

A marriage lived out in faith and authentic love is hence the greatest protection against abusive relationships that we have.

How this squares with abusive relationships

What happens, however, when—because of the free choices of one or both of the spouses—there is a breakdown in marital love? What happens when one spouse begins to abuse the other?

It is important to realize that authentic Christian love does not oblige a person to endure dangers to his own safety or to the safety of others. On the contrary: grace and supernatural love exist in order to protect the integrity of the person. If a person is living in a situation that is objectively and gravely harmful to himself, or to those under his care, then he is certainly permitted, and may even be obliged, to flee from it.

There are extreme cases in which a person is permitted to risk his life or some other grave harm, but they occur only when some other, much greater good is at stake. For example, a person offered a choice to renounce his faith or suffer martyrdom is justified in choosing to suffer martyrdom (CCC 2473); similarly, because of the imminance of the the danger, a person may choose to risk his own life in order to save someone else in danger of drowning.

However, remaining in an abusive relationship is not one of those cases. Although marital unity is a great good, it has already been gravely transgressed by the offending spouse. By leaving such a situation, the injured spouse is not instigating a new rupture; he in only acknowledging one what already exists. (Separation, in any case, does not dissolve the marriage.)

Conclusion

Based on the discussion above, I will now answer the O.P.'s questions succinctly and address his concerns.

It is true that, because of Original Sin, we are unable to love as we ought. However, we must not underestimate the power of sanctifying grace, which we receive at Baptism (and the grace that spouses receive in the Sacrament of Matrimony). Those graces actually restore our ability to love as we ought. Moreover, it is our doing: when we love authentically, it is Christ who loves in us.

Authentic Christian love does not consist as such in emptying ourselves or giving in to others. Rather, it consists in allowing ourselves to be full of Christ, and giving him to others. We have nothing to give to God or others, unless He first gives it to us.

Remaining in an abusive relationship is absolutely not the same thing as martyrdom. With martyrdom, one’s life is weighted against a vastly greater good: namely, the Faith. However, those remaining in abusive relationships (although evidentlywe must never judge them harshly) are choosing to endure sufferings for the sake of a good that has already been, to some degree, ruptured. If the abuse is grave and sustained, they have every right (and possibly an obligation) to flee from it.

Although the abused spouse is entitled to flee from the situation, he is still called to love even his abuser. However, love must be understood here to mean supernatural love, not the feeling of love. Supernatural love entails desiring what is best for the beloved, and it does not necessarily entail feelings of any sort. No one blames the victim of abuse for having feelings of disgust or even hatred toward his abuse, but he is nevertheless called to rise above those feelings. Authentic love will lead him even to forgive his abuser.

Note that forgiveness does not mean returning to an abusive spouse (and, in fact, in most cases, returning would not be advisable). It does entail desiring what is best for him: his repentence and conversion, as well as his eternal salvation.

Finally, I think it would be mistake to say that those who remain in abusive relationships are committing idolatry. Although they will not find God, certainly, in the abusive behavior of their spouses, neither are they trying to offend Him. Abused spouses suffer terribly, and so they are not really in control of their situation. They are badly in need of help (if possible, professional counseling). In other words, they tend to remain in the relationship for psychological, rather than moral, reasons: it is not so much their free choice, as their dependence on the other, that keeps them there.