If a person is ever cloned, does that living being have a soul? Can we say that God made that person, and that despite being cloned by man, is still a unique individual? Should a clone of a person have the same rights as a person who is, well, the original? Would it be a sin for a person to allow themselves or to allow others to be cloned?
The following is my response with some additions:
To answer your question: every human person has a unique, unrepeatable soul, no matter how he/she came to be. The means by which a human clone came into existence would be evil (assuming that scientists are actually able to figure this out, and one day do), but the person him/herself would still have all the rights as anyone else (just as a person who was conceived in a test tube or in rape would have been conceived in an evil way, but would still have his or her own unique soul with all the rights as anyone else).
If a human is alive, he must have a soul, and souls only come from God. St. Thomas Aquinas wrote:
the soul is defined as the first principle of life of those things which live: for we call living things "animate," [i.e. having a soul], and those things which have no life, "inanimate." (Summa Theologia I.75.1)
the human soul is produced by God (Summa Theologia I.75.6)
Like all moral situations God gives us free will to do things the right way or the wrong way. Rape, artificial insemination, sex outside of marriage, and cloning are all wrong ways of bringing a human being into existence. God also allows all the consequences of our actions (good or evil), so He just might allow these evil situations to be effective in their natural consequences (creation of a new human life) by creating a soul to animate the body these people are forcing into creation--even though these means are against His design.
Yes, you're right. People who practice things like artificial insemination and cloning are (knowingly or unknowingly) trying to be as great as (or greater than) God. They're taking a place that belongs to God in the human reproduction process. Every human being has the right to be the fruit of the loving act of his/her mother and father (c.f. CCC 2378). God designed humans to be the natural result of a loving marriage, not the result of scientific processes. Any act that creates a human being outside of the marital act, then, is evil (contrary to God's design). It would also be a sin for a person to willingly have him/herself cloned (for the same reasons).
The only exception for this is, obviously, natural cloning (a.k.a. twins). If I remember correctly, I think I heard somewhere that identical twins (or was it fraternal twins?) are considered clones of each other because of the way they develop in the womb. That's something about which an embryologist would a have much better understanding, so don't quote me on it.
Some might try to argue that if clone has a soul, and God created the soul, this must be proof that God thinks cloning is okay. Again, we refer back to the fact that God allows the natural consequences of all of our actions (good or bad). If I jump off a tall building, God will let me die (no matter how good or evil of a person I am) because death (separation of the soul from the body) is the natural consequence He has set up for things like falling from high places. If I steal food from a restaurant, it will still taste the same as if I had bought it honestly. God would not be allowing it taste good because He condones my theft; rather, He would allow the experience of the good taste of the food to be a natural consequence of putting good tasting food in my mouth. Too often, people try to excuse their evil actions because they do not see an immediate reaction by God (like lightning bolts shooting out of the sky to kill them for doing whatever they did). They're just fooling themselves into thinking their activity is okay.