According to some scholars, existentialism posits that there are motivations which move us to action that we cannot understand and may not be aware of at all. This would seem to jive with the possibility in being an "existentialist" while not believing in free will.
This is open to debate however.
The nihilism question is a bit tougher, or seems tougher to me. Nihilism and existentialism are both rather loosely defined because, as terms, they are tossed around quite a bit and applied in contexts that widen their meanings. Nihilism suffers more than existentialism in this way.
My understanding of nihilism, if we take it as an absolute philosophical principle, is that under nihilism there can be no profound meaning of any kind, personal meaning would be included. The world was not born meaningfully, does not possess or express an intentionality which would be required for "actual meaning" and so offers only contingent, momentary, arbitrary and accidental meaning.
Of course, words still have meaning. Actions still have meaning. But the meaning stems from a socially-codified agreement, not from a pre-existing, larger context. There is no reference frame outside of human history and so there is no objective or spiritual meaning available. It's all just people, people's attitudes (inherited and imagined), and that's it. That's nihilism, as I understand it.
Importantly, there is no call for destruction, for immoral behavior or any negativity at all as a natural consequence of nihilism. Essentially, both existentialism and nihilism share the notion that there is no ultimate reference frame from which we might derive meaning. Existentialists say that if it's there we can't know it. Nihilists say that it isn't there at all.