In the First Meditation, does the Meditator want to suggest the possibility of a universal dream or the universal possibility of dreaming? In other words, is he suggesting that all life might be one big dream or simply that we could be dreaming at any given moment for all we know?
There is no definite answer to this question, and it is debated among interpreters. Perhaps the interpretation more consistent with Descartes' wider project is the universal possibility of dreaming. We could see this idea, this measure of doubt, as meant to question the Aristotelian reliance on the senses without doing away with knowledge and the world altogether. If he were suggesting the possibility of a universal dream, the Meditator would be sweeping away a great deal more than just Aristotelian epistemology. Also, the Painter's Analogy which follows the Dream Argument seems to rely on the fact that there are things in this world that we can derive images from, which would suggest to us that the Meditator has not yet fully abandoned the notion of a material world.
What stops the doubt of the First Meditation? What kind of reasoning supports the cogito?
This crucial question is infuriatingly difficult to answer. While the classic formulation of "I think, therefore I am" is easy to read as a syllogism, that reading is probably inaccurate. After all, it comes at a time when the Meditator has cast even rational thought into doubt. More likely, the cogito is meant as an intuition rather than an inference. Part of the puzzle lies in the fact that the Meditator calls the cogito a "clear and distinct perception," but then goes on to suggest that we can only be certain of our clear and distinct perceptions once we have established that God exists. If that is the case, then the cogito is not confirmed at all until a bit later in the Meditations.
What does the Wax Argument show? What is it meant to show? Does it succeed?
The Wax Argument is meant to show that the mind is better known than the body. It does so by suggesting that everything "I" know about bodies "I" know through intellectual perception rather than through the senses. Since every act of thought reinforces the cogito that also suggests that "I" am a thinking thing, every act of thought brings me closer to understanding my own mind. We might question how accurate this assessment is, however. Every act of thought may reinforce the cogito, but that doesn't mean it brings me closer to an understanding of my mind every time. It just reinforces the same one piece of knowledge--that I exist. But perhaps Descartes is not thinking of items of knowledge when he says that the mind is better known than the body. Perhaps he simply means that it is known more distinctly, and a constant reinforcement of the mind's existence might help to give the distinct knowledge.