Are Cantor's set theory and the axiom of choice well accepted nowadays? Is there a "more consistent" theory ? Do mathematicians agree about the validity of the results obbteined by this theory ?
 A: I don't think the question is unclear. You want to know if most mathematicians still work with set theory, roughly as formulated by Cantor, and you want to know if most mathematicians worry about issues raised by using the axiom of choice. 
For any reasonable definition of "most," the answer is that most mathematicians still work with set theory, roughly as formulated by Cantor, trusting that some or other formal codification of this is consistent, and that most mathematicians do not worry about the axiom of choice.
A: In my experience, I never seen a single mathematician even mention the Axiom of Choice during lecture/talk. If they choose elements in every open cover, then they are perfectly okay with that, and never even make a big deal about it. But after two months of choosing elements in open covers, they decide to give a proof of Tychonoff theorem, and announce "today we will use Axiom of Choice in the form of Zorn's Lemma". It is as if they never even realized that they been using Choice for the past two months. 
So the short answer is, most mathematicians use choice all the time without even realizing it (unless they are logicians). They only mention it when they form it in the form of Zorn's Lemma or Well-Ordering Principle. 
The classic joke describes this situation perfectly: "The Axiom of Choice is obviously true, the Well Ordering Principle is obviously false, and Zorn's Lemma is a nice technical result to use in a proof".   
