May I choose to reenter the U.S. using my valid H1B visa rather than using the Advance Parole ?

Yes you may infact reneter using your H1B, All the best.

How could we search the processing time of MTR in Vermont centre in USCIS website

You will not be able to find out through the website, you can call customer care representatives to find out. All the best.

How could we search the processing time of MTR in Vermont centre in USCIS website

You will not be able to find out through the website, you can call customer care representatives to find out. All the best.

My Question is: will it affect also the I-765 ( Work Permit) as the letter just mentioned the I-485 case and I had my biometric

Most likely yes, try to send all the documents required as soon as possible. All the best.

You should go ahead and file with your married name, USCIS will know they process a lot of these applications every day. All the best.

What can I do to handle the situation if you foresee a problem?

If you can change one of the two (i am guessing passport) do it and then file for EAD. Even if you file I do not see a big problem as long as you...

This wouldnt be a problem, you do not have to wait for 2 weeks. All the best.

How do I qualify under this rule to become a US citizen.

You might not be able to get US citizenship directly. And you just got off probation you might have to wait 5 years before applying for US...

New OPT STEM Extension rule

You will get one OPT for bachelors and one for Masters, you may not be able to get another OPT if you do another masters.

For case specific advice it always better to contact an immigration attorney in private rather than this public forum. All the best.