The second part of your explanation is extremely enlightening and made me understand with some depth the meaning of the expression "smoothness spreads out from the generic fiber".

Dear aonymous, thanks a lot for your answer. I had completely forgotten about the Nullestellsaz (and elemination theory), but now I remember Wan der Waerden wonderful treatment in his second algebra book. I really like this kind of argument since it is extremely elementary and it goes directly to the heart of the matter without burying it into fancy words which sometimes hide the essential truth.

Thanks @Felipe. You said in your answer (in the reference link you gave) "...with schemes it is completely obvious...", so may be you could write down or give me a sketch? Thanks in advance

@Ari, is it possible to prove this result using the notion of resultant combined with an decreasing induction on the dimension?

sorry sorry, I meant smooth connected (so irreducible), I'll reedit it!

Yes Q1 is actually motivated by a problem of one of my colleagues, and yes, most likely he won't need to hold an explicit uniformizer. Nevertheless, I thought the question so natural that I could not resist to post it on MO.