Are there inherently ambiguous and deterministic context-free languages?
If a language $L$ is deterministic, it is accepted by some deterministic push-down automaton, which in turn means there is some LR(1) grammar describing the language, and as every LR(1) grammar is unambiguous, this means that $L$ cannot be inherently ambiguous. Knuth proved this in his paper in which he introduced LR(1) (On the Translation of Languages from Left to Right).

A language can be described by some context-free grammar if and only if it can be recognized by some nondeterministic automaton. As a special case of this, inherently ambiguous context-free grammars can be parsed by some nondeterministic automaton.

On a final note, any deterministic push-down automaton is also nondeterministic (this is the case for just about anything that can be nondeterministic, for a reasonable definition of nondeterminism).