Ambiguity of regular expressions Some regular expressions are ambiguous. Some are not. a*b* is unambiguous for example. Expression a*a* is ambiguous but it can be written in the unambiguous form 'a*`. The answer to this question gives an algorithm for deciding whether a regular expression is ambiguous.

Is there an algorithm for finding an equivalent unambiguous form of any given RE?
Are there REs that are inherently ambiguous?

(This question seems relevant by title; not by content)
Yes, every regular expression can be converted into an unambiguous one by converting to a DFA and then to a regular expression. And no, there aren't any inherently ambiguous regular languages in the sense described in the question. This is a classic result in automata theory:
 R. Book, S. Even, S. Greibach and G. Ott, Ambiguity in graphs and expressions, IEEE Transactions on Computers 20(2) (1971) 149–153. 
See also this question over at MO for more details and a reference: https://mathoverflow.net/questions/45149/can-regular-expressions-be-made-unambiguous