Let me ask you again about what I can do without a social security number. If I need to get a driver's license, can I actually get a social security number?
no, if that's your only purpose for asking a social security number, then no, you can't.
In case I change my name, do I have to inform my employers?
You should, yes. If you don't it could prevent your wages from being posted correctly to your Social Security record and that would lower the amount of your future social security benefits. It could also cause delays when you file your taxes.