If you are in the U.S., you cannot work. Period.

It could be an individual or a company responsible for sponsorship.

As long as the application was timely filed, it does not matter what the notice date is.

It is better to strategize with assistance of an attorney rather than seek general information which may or may not be suitable to your unique...

Since you are self petitioning, the employer is irrelevant

You are not maintaining H-1B or any U.S. immigration status when abroad

Consider studying along with her and get your own F-1

There is no requirement to be employed by petitioning employer prior to or during GC process

You will benefit from an approved H-1B as a long term plan.

Not sure why they cannot seek it in your behalf.