Here is how I would have answered these question, if I were taking the test.  These are not the only correct answers.  (I omit definitions, since you should have those in your notes.)

1. The Third Cosmological Argument inspired by St. Thomas Aquinas contains the following key premise:

5. If I exist now, and I am a contingent thing, and every contingent thing that exists has a cause, and causes precede their effects, then there was a non-contingent first cause.

(a) Define the two technical terms in this premise.

Something is contingent if it exists but it might not have.
Something is non-contingent if it exists necessarily (i.e., it could not possibly have failed to exist).

(b) Provide the rationale for this premise (i.e., explain how Aquinas would reason from the first half of the premise to the claim that there had to be a non-contingent first cause).

If these four things are true, then there has to be a series of causes that came before me leading up to me.  Such a series must begin somewhere; it must have a first member.  This first member is thus the first cause in this series of causes.  This first cause must be uncaused, because we know it's not self-caused (given our supposition that causes precede their effects) and we know there is nothing that exists before it to cause it (since it is the first cause).  Since it is uncaused, it must also be non-contingent; this follows from our supposition that all contingent things have a cause.  Therefore, if these four things really are true, then it follows that there was a non-contingent first cause.

(c) How would Leibniz criticize this premise?

Leibniz thinks that it is possible for the world to be eternal; that is, for it to have no beginning and no end -- just an infinite series of causes and effects.  But if the world has no beginning then it has no first cause.  So it might be the case that these four things are true (i.e., that I exist now, and I am a contingent thing, and every contingent thing that exists has a cause, and causes precede their effects) without it being the case that there was a first cause.  The causal chain just goes back forever.

2. Here is Leibniz's Argument from Sufficient Reason:

1. The world exists.
2. If the world exists, then there is a reason why the world exists.
3. If there is a reason why the world exists, then there is something outside the world that the world depends upon for its existence.
4. If there is something outside the world that the world depends upon for its existence, then there is a necessarily existing, Creative being.
5. If there is a necessarily existing, Creative being, then God exists.
6. Therefore, God exists.

(a) E.E. Leibniz's Argument from Sufficient Reason. If you said in the 'Evaluate' step that you think the argument is sound, now discuss what you take to be the strongest criticism of the argument.

Explain

P1
Technical Terms:
world
Rationale:
I know I exist.  This is enough to know that the world exists.

P2
Rationale:
This follows immediately from PSR, which says that for everything that exists, there is a reason why it exists and is as it is rather than some other way.

P3
Rationale:
If there is a reason why the world exists, then, since the world is a contingent thing and since contingent things can only be explained by things outside of them, there must be something outside the world that explains why the world exists.  Since this thing explains why the world exists it is fair to say that the world depends upon this thing, whatever it is, for its existence.

P4
Technical Terms:
necessarily existing, Creative, ontologically dependent
Rationale:
This premise follows simply from the meanings of 'necessarily existing', 'Creative', and 'world'.  If something is outside the world, then it must, by definition, be necessarily existing, since the world is defined as the collection of all contingent things.  If the world (i.e., all contingent things) depends upon this thing for its existence, then we can also infer that this thing is Creative, since for something to be Creative just means for all contingent things to be depend on it for their existence.  Thus we can see that if there is this thing outside the world that the world depends upon, this thing must exist necessarily and be Creative.

P5
Technical Terms:
God, omnipotent, omniscient, omnibenevolent, incorporeal
Rationale:
If there is this thing that has these two pretty remarkable divine attributes, then it seem reasonable to infer that this thing is God.

Evaluate

1. Valid
2. MMP
3. I think this argument is unsound.
4. I think it is unsound because I think premise 3 is false.  I think premise 3 is false because I deny PSR.  Why can't there just be "brute facts" -- facts with no explanation whatsoever?  The world just exists, as I see it, and there's nothing that explains why it exists.  There is no explanation for the fact that there is something rather than nothing.  

3. (a) Explain in a paragraph or two what the Problem of Evil is. (Be sure to say what claim is being argued for and what the reasoning is.)

The Problem of Evil is this: certain facts about the world seem to be inconsistent with the existence of God, as theists usually conceive of him.  God is defined as omnibenevolent and omnipotent.  Thus, he can bring about any state of affairs possible, and he always prefers the best state of affairs possible.  Well, the best possible state of affairs surely would contain no evil.  So, it seems that if God exists, then there is no evil.  But there is evil (women are raped, children are abused, animals suffer -- in short, there is needless suffering).  So, because there is evil, it seems to follow that an all-powerful and all-good God does not exist.

(b) What do you think about the Problem of Evil? Does it refute belief in a supreme being once and for all, or does the theist have an adequate reply? Discuss.

I think the problem of evil is a very strong argument against theism.  I grant that God might have a good reason for allowing some pointless suffering to exist, but certainly not all of it.  For example, I heard a news report this year about some deer that died in a forest fire in Yellowstone.  The deer were severely burned and then laid there, suffering and slowly dying, for days.  The suffering served no purpose, and it was not caused by agents exercising their free will.  It was caused by natural events.  Even if there was a purpose for the forest fire, certainly there was no reason that the deer had to suffer for days instead of dying immediately.  Since the way things turned out was not the best possible way for them to turn out, I don't see how there can be a being that always prefers the better to the worse and has the power always to bring about the better.