The U.S. has no rules against a person working outside of the U.S., so yes, if you get an H4 and an EAD you do not need to live in the U.S. and you...

It seems to me that you could argue that you did not engage in employment by receiving a cash prize.

If your U visa is based on being a unmarried child of a U visa applicant, then getting married can affect your eligibility.

Regarding the reason for extending the stay, you do have to have a reason, which you will have to articulate and prove. Regarding the return...

If you are single and never married you should indicate that you are single and never married. Yes, it may impact your eligibility for a visitor's...

Call them or email them, or get a supervisor somehow. This is common, and requires the intervention of a human, not an AI program.

Yes, you can file an I-140 even if you are trying to consular process for your DV1 visa, there is no rule against doing so.

It is completely at your discretion which passport you wish to present to the medical provider for your medical exam.

If his green card just recently expired you should file an I-751 for him and he should enter on his expired green card and I-751 receipt.

Check the I-751 receipt to determine if it is expired and you need a stamp in order to travel. Otherwise, it is safe to travel.