What are some of the reasons for which Catholics are against gender-neutral secular marriage?

Have you heard of the natural law? It is an rational ethical system based on the "ends" or purposes of human nature. We are against any government that rejects the natural law, as the natural law can be known by human reason alone, so any rational person should accept it. It is one thing for the secular state to reject religious law; it is another for the secular state, which glorifies reason, to reject the natural law which can be known through reason.

Following Scripture and the Church Fathers, as well as the Natural law, Catholics understand sex as being for the purpose of procreation. In fact, the denial of procreation, either as itself (contraception, sodomy, masturbation, etc.) or its effects (a child that needs a family to raise him, which divorce frustrates) are the cause of all merely sexual perversion.

Notice, from the Catholic point of view, that if you accept contraception, we deny that you can argue against sodomy. If you don't think sex is for procreation, which approval of contraception requires, then you can't denounce sodomy either, except of course by appealing to Divine Command. We actually agree with the author of the second article that, given marriage is not about procreation, it follows that there is no reason to reject gay "marriage." However, we Catholics simply deny the premise "Marriage is not primarily about procreation." To put it another way, we think that the acceptance of contraception (which all Christians, Protestants and Catholic, didn't accept until the early 1900's) led to the acceptance of sodomy.

Anyway, one might notice that rape is not really against procreation in itself. I wish to clarify that the evil in rape is not in the sex act, but rather because the sex act happened without consent. As such, this makes rape far more evil then any of the sins above, because instead of it being a sin against sex, it is a sin against justice.

Modern sexual ethics, on the other hand, don't accept that sex has any inherent purpose, but rather a purpose as we define it. In other words, sex can be for procreation, but only if we choose for it to be so. From this principle, no sex act alone can be wicked; only sex without consent can be wicked, as rape is not evil because of the sex act, but rather because of non-consent. This is why I mentioned above how rape is not a problem of procreation, but a problem of consent- and justice.

In fact, the entire principle of the Sexual Revolution is that, in the words of Philosopher Peter Singer "sex brings up no serious moral questions." Any sex act*, under this view, is good, as long as it is consent among all persons involved.

Christi pax.

*Pedophilia is wrong, in the Sexual Revolutionist's eyes, because a child is not mature enough to consent.

Catholics believe that one of the duties of the secular state is to uphold moral law. Secular authorities should make things that are bad (like theft, fraud and tax evasion) illegal. They believe that people benefit from following moral laws, even if they don't believe in the source of those laws (God). They also believe that this makes for a better society.

Pope Leo XIII's encyclical on marriage, Arcanum §23, gives a succinct answer to your question:

23. Let no one, then, be deceived by the distinction which some civil jurists have so strongly insisted upon—the distinction, namely, by virtue of which they sever the matrimonial contract from the sacrament, with intent to hand over the contract to the power and will of the rulers of the State, while reserving questions concerning the sacrament of the Church. A distinction, or rather severance, of this kind cannot be approved; for certain it is that in Christian marriage the contract is inseparable from the sacrament, and that, for this reason, the contract cannot be true and legitimate without being a sacrament as well. For Christ our Lord added to marriage the dignity of a sacrament; but marriage is the contract itself, whenever that contract is lawfully concluded.

His argument is, in a nutshell, that the Church has jurisdiction over marriage, not the State, because marriage is sacred and that is the realm of the Church. In other words: there is no such thing (at least for the baptized) as secular marriage divorced from sacramental marriage. Just because the State benefits from marriage, this does not mean it has lawful authority over it.

Also, to call marriage what is not marriage is contrary to objective truth and the Catholic understanding of human law as flowing from and being in-line with natural law and divine law. "The Canonical Nullity of the Marriage Process as the Search for the Truth" by Raymond Leo Cardinal Burke (the last article of the anti-Kasper Remaining in the Truth of Christ) says:

Regarding law, in general, there has developed the notion that the law has no relation to objective truth but is constituted by whatever man, usually the judge, decides. Such a theory was proposed in my homeland, the United States of America, already in 1897 by Justice Oliver Wendell Holmes, Jr. [The Path of Law and the Common Law pp. 1-29]