1) She is bit shy though.
You can't use "bit" without "a."
2) She is a bit shy though.
3) She is a bit of shy though.
You can't say she is "a bit of shy," because "shy" is an adjective. "A bit shy" means "slightly shy." "A bit of shy" would mean "slightly of shy," which doesn't make sense.
4) She is a bit of an introvert.
This is a slightly different construction. The sentence means "She is somewhat of an introvert." You use "of" here because "introvert" is a noun.
Contrast these constructions:
"She is shy." "How shy is she?" "She is a bit shy."
"She is an introvert." "How much of an introvert is she?" "She is a bit of an introvert."
Hmmm...that's actually exactly what I was thinking. But you can say: 1. What kind of creature is that? (without determiners: a or the) 2. That's kind of weird. (of + adj., which is against your rules) What do you think?