Question 3

Luck has nothing to do with it.

The statement above is a sentence fragment.

The statement above is a complete sentence.

Question 4

The statement is a complete sentence.

Explanation:

The question is, do you need more information to understand what is said?

I think the sentence is just fine. You get the idea of what it means immediately.

For example if you had a sentence that said

Because I practiced throwing a curve ball, I struck the batter out.

and then you followed it with

Luck had nothing to do with it.

You should understand that I got the batter out, and luck had nothing to do with it.