1. If my new employer will sponser my H-1B visa, when is the latest time they need to the petition to USCIS?
-No latest time. Technically, your H1 status expired when you lost your job. But if your new employer can file your H1 petition within one to two months, you have a good chance to get it approved with I-94 because USCIS usually does not so strict.
2. If I don't have my H-1B transferred to a new employer within the grace period, what should I do to maintain my status.
-Your status is AOS now, so you do not have status problem.
3. My NIW petition was not sponsored by my employer, am wondering if I am OK to stay with unemployment and how long.(I am currently holding a valid EAD though I never used it in the past.)
-Yes, as long as your I-485 is pending, you are OK.