Connotative differences between Spanish "cara", "rostro", "haz"?

Factual Questions
1

I am assuming that while the Spanish words cara, rostro, and haz are usually translated into Eglish as “face”, their usages in Spanish do not completely overlap.

What are the connotative differences between the three? Are they all usable to mean “the human face” in all contexts, or are there some usages which sound odd to native Spanish speakers (e.g. maybe OK to say “She slapped mycara”, but odd to say"I’m washing my cara")?

Or is this one of many, many things that break differently in different parts of the Spanish-speaking world?

2

It’s faz, not haz.

I’ve said “lavarme la cara”. I don’t think I’ve ever paid much attention, but by the dictionary, it seems rostro is more restricted to human faces, while cara can also be used for animal faces.