All deonds.

1. In general, yes. But if you work for no-profit organization first, then you want to work for profit company you are under quita cap.
2. Yes.
3 & 4. Once you quit your job before you find a new one, you are out of status when your current employer report your leaving the job to USCIS. There is no grace period for H1B. It all depends on the immigration officer's discretion to decide that.