You should use have/has got to say that someone has a particular thing, or to mention a quality or characteristic that someone or something has.
I've got a coat just like this...
She hasn't got a work permit...
However, In informal American English, people sometimes just use `got'. (SPOKEN).
2nd Question:can I use "screen" to refer to "computer display"?
Yes, you can. But you cannot say out of sudden, "I've got a new screen" because others won't know the context (what kind of screen; TV, car etc.)
So I would like to suggest you to say computer (screen/ monitor)...
"Hey, I've got a new screen for my computer"
PS: Not a teacher.