If filed by an agent, they can represent a new company. If it was directly filed by employer, then the new employer files their own petition.

It is not mandatory, however you should not wait to file. If the application is not properly filed, it could get kicked back leaving you out of...

It is not necessarily a problem, however prior marriage is considered and you may be interrogated for bona fides of the filing.

What are other alternatives for a person whose on L1A to start the GC process when Employer is mot willing to start soon? Look for other options...

If H1B is pending and the employer informs the USCIS of the lay off, USCIS will discontinue processing.

Job for future offer, upon acquiring residency, it must be available. If not, you risk getting a NOID and getting the residency revoked.

Not sure why you are considering being subject to the lottery if you have I-140 approved and extensions possible until PD is current

Yes, it appears to be unauthorized work and violation of status. You should not continue to stay and work in the U.S.

Both are non immigrant visas so from a burden point of view, both will require you to show non immigrant intent

If your priority date is current and you are maintaining status, you can apply for AOS provided you did not just enter the U.S.