Can I work on OPT (initial 12 months) with a non e verified company? And how long does it takes to get a company e verified?

The company has to be a e-verified.

Can a permanent resident get married to someone on a non-immigrant visa ?

You should sponsor your girlfriend using a proper immigrant petition. Only spouses of US citizens will be forgiven for overstaying their visas...

What all complications will I face in the above mentioned case? Can I maintain a domicile of USA while I am abroad as mentioned?

You need to make sure you will not lose your GC status. You should consider filing for reentry permit.

Form I-765, qs 16 Eligibility category

Are you filing for your spouse based on marriage? you can look up categories in the instructions to the form.

You cannot misrepresent information. If you re asked if you are married, you have to answer truthfully. As for the rest of your questions, you...

The only way to restart the clock in this situation would be to appear for a scheduled interview. You need to file a request to schedule your...

Question on H4 to H3

You can check USCIS processing times on their website. H3 and H4 are two different things. There can be many different issues during the change of...

Will I eligible for cap-exempt through employer B. Kindly confirm?

Employer B will have to start the process again. Good luck!

Asylum case pending and i got married

You should be able to adjust your status based on marriage. After that, you can ask USCIS to dismiss your asylum claim.

One who is a convicted of a particular serious crime is not eligible for asylum. You need to work with good lawyers.