Yes, you apply for H4 EAD on H4 with reopened status.

The appropriate answer is "yes" because you were refused a visa.

Yes, if stayed longer than annotated on I-94 by CBP officers.

A USC with DUI is not a barrier to sponsor applicant for US residency.

Yes, apply for family based green card while employment based is pending.

Yes, you may travel abroad with a valid F-1. To avoid problems on reentry, go to the US embassy consulate for a New F-1.

The EAD is adjudicated before the extension of the H-1B.

Yes, secondary applicants start using GC EAD and primary applicants can stay on H1B.

Yes. A separate I-864 is required for each beneficiary.

No. This question was addressed yesterday. Check the threads..