Is a universe where absolutely nothing exists conceivable without contradiction?

It does not seem inconceivable to me: especially if one draws a distinction between a universe and the objects in it, it certainly seems conceptually possible. And a brief search on the web suggests that this conclusion is not merely the result of uninformed, armchair speculation: click on this link.

It seems that many philosophers use the "socrates" argument to explain a simple deductive argument. This argument is P1: All men are mortal P2: Socrates is a man C: Therefore, Socrates is mortal. However, is this not begging the question because P1 assumes that Socrates is mortal?

An argument like the one that you presented, which has the form: (1) P; (2) P-->Q (i.e., If P, then Q); Therefore, (3) Q, if introduced in order to explain a simple deductive argument, is meant only to illustrate the concept of soundness. An argument is sound if the conclusion follows logically from the premises. But a sound argument need not be valid, or yield a true conclusion. An argument is only valid if it is sound, its premises are true, and the conclusion follows non-circularly from the premises. Although the argument that you present is--at least to the best of my knowledge--not only sound, but valid, the two can of course come apart. So, for example, the following argument: (1) The sun always shines in Southern California; (2) Los Angeles is in Southern California; (3) Therefore the sun is always shining in Los Angles is valid, it's not--_mirabile dictu_ sound.

What makes an argument "good"? Is there more to a good argument than raw persuasive power? Does a good argument have to support the right conclusion? For example, might the ontological argument be a good argument for theism even if theism is false?

This is a deep and interesting question, which goes to the heart of what exactly the point of philosophy is. Let's begin, however, by fixing some ideas about arguments. Arguments may be either valid or invalid; sound or unsound. An argument is valid if and only if the conclusion of the argument is a logical consequence of its premises; an argument is sound if and only if it is a valid argument with true premises. So, for example, the following argument is valid: (1) All green ideas sleep furiously; (2) This idea is green; (3) Therefore, this idea sleeps furiously. (This example is derived from Noam Chomsky, who used the phrase 'Colorless green ideas sleep furiously' in order to illustrate that competent English speakers could recognize a meaningless sentence as grammatical.) Obviously--I would claim--the preceding argument, although valid, is not sound, because ideas aren't colored and they don't sleep (except in the most metaphorical of senses). By contrast, the following argument is both...

What does the term primitive mean in logic? Is it something predicated to an item or concept to denote that this item cannot be any further explained or reduced to still more concepts?

Primitive terms in logic are, indeed, those that cannot be defined further, they are basic starting points--like axioms in Euclid's geometry. It seems to me, however, that questions can and indeed are raised about the nature of these primitives. Are they necessary truths? Are they simply necessary relative to some system? Such questions continue to be investigated by logicians and philosophers of logic.

According to Descartes' demon hypothesis, would it be possible for the demon to deceive us about the rules of logical inference e.g. could my belief in the law of non-contradiction be caused by the demon?

Jay is correct that eternal truths are up to God. In a letter to Mersenne, Descartes says that "since God is a cause whose power surpasses the bounds of human understanding,and since the necessity of these truths does not exceed our knowledge, these truths are therefore something less than, and subject to, the incomprehensible power of God." Nevertheless, Peter is quite right--in virtue of the textual evidence that Alex cites--to say that the evil deceiver (or 'omnipotent God') doubt is introduced in order to cast mathematical truths into doubt. It's worth noting, however, that in the Third Meditation, this doubt about eternal truths is characterized as "slight and metaphysical". Indeed, Descartes writes: "Yet when I turn to the things themselves which I think I perceive very clearly, I am so convinced by them that I spontaneously declare: let whoever can do so deceive me, he will never bring it about that I am nothing, so long as I continue to think that I am something ; or make it ture at...