Is 'a black shoes' grammatically correct?

Answer 1

No.

The indefinite article "a" comes before singular nouns, and "shoes" is plural.

In the right context, you could use "the". Example: These are the black shoes that I've always wanted.

Depending on the sentence, you might not need an article at all. Example: I need black shoes to go with my dress.

Answer 2

If you're asking a question, yes.

For example,

"Ethan bought some shoes."

"What color shoes?"

You need a question mark at the end.