Your employer is most likely not permitted to require you to reverify your lawful permanent resident status. Regarding travel, though, certainly...

I expect CBP will indeed flag your name if your mother was found to be carrying your passport into the U.S.

If you were paroled into the U.S., yes. If not, perhaps you qualify, but simply being released from detention, by itself, does not qualify one for...

If she was granted a change of status to H1b and she is not working for the company, she is in violation of her status.

No, only a spouse or minor child can get an h4.

She cannot file for naturalization, and she is not a U.S. citizen by birth or naturalization, or as a derivative. As such, she can file when she...

Could be. Try contacting CIS customer service.

You should follow up with them. Perhaps send a copy of the I-485 and receipt notices, and a formal motion to reopen.

The risk is that your company may have a very good reason to want to refile the I-140 and CIS may deny your 485 if there are issues with the...

You are not required to get a new visa abroad at all in order to port to a new employer.