>Question 1: Is there a classification of (noetherian if needed) integral domains with finitely many units ? (of course we can exclude fields as trivial examples)

Probably there are many such domains that I can not think of but maybe there is an overview article of what is known.

>Question 2: Is there a classification of principal ideal domains with finitelym any units?

This seems to be much more restrictive. Examples include the (finitely many) imaginary quadratic fields with class number one and $F_q[x]$ for finite fields $F_q$. 

If it helps one can also assume some additional things like characteristic 0.