Welcome to the forum, you will find much useful (and some useless) information here
Screws should not be popping, sounds like it is not fixed properly, Did they remove the old ceiling or just sheet over it? The screws that are popping more than lilely are not screwed into timber and need to be removed and longer ones put in.
I'm no plasterer but I have fixed a few ceilings that have sagged with 100% success, so it ca't be all that hard.
Get them back to fix it.
I wouldn't be paying until I was happy with the result.
Good Luck
RegardsBradford