Further, do monads warn the user when there is an ambiguity in the "grammar"?
Monadic parsers can do context-sensitive things that context free parsers can't, but they can't have unordered options which context free parsers can. So really monadic parsers have different powers to context-free.
> do monads warn the user when there is an ambiguity in the "grammar"?
You don't have ambiguity in the grammar because monadic parsers only provide ordered options.
Oh, but you do have ambiguity in the grammar, except that parser returns one of the possible parse trees deterministically and thus doesn't warn you that the input could be parsed differently. This leads to the false impression that your grammar is unambiguous.
Specifically, it makes it impossible for you to even write down that ambiguous grammar you wanted.
I'm not aware of a parser combinator library that does this. The way you build parser combinators in Haskell doesn't exactly preclude it from being done but it wouldn't be totally trivial either.
Googling for it didn't turn up much that wasn't a veiled reference to this paper:
http://richard.myweb.cs.uwindsor.ca/PUBLICATIONS/PADL_08.pdf