The phone snatching is a red flag for sure! I absolutely would not tolerate that bullshit.
He is obviously lying about his job title and income to make himself feel better. That kind of insecurity is a major red flag! And the fact that he is talking about marriage and children when he can't even be honest about his job is a sign that he's trying to trick you into getting stuck in this relationship before you know the truth about him.

You should gently confront him, tell him that if he wants a future with you he needs to stop lying and build a relationship with honesty and trust. Give him a little time to figure out if he can do that and be ready to leave if he doesn't.