What are the option for me to prove my legal employment during 2010-2011 years? Where and how I can retrieve old EAD?

You will need to provide evidence that you had the J-1 visa.

Mavni husband

The benefits do not extend to the spouse. However, once your husband acquires status he can apply for you.

How can my husband stay without doing voluntary departure

No there is very little that can be done now that the judge has ordered removal. The only thing is an appeal.

I have a strong ground for asylum, did I miss the 1 year deadline and can I still file it ?

You do appear to have missed the one year deadline but there are exceptions. Apply anyway. Discuss with an experienced immigration first though.

Simple answer is yes you will be able to apply for OPT again in conjunction next time with the new master's degree.

You can file the I-290 and explain you never received the RFE. Which is better? Usually the one that does not cost so much money.

Gap in job for 40 days

That gap of 40 days can be very detrimental. Discuss this with the attorney for company B.

What is the process of sending additional documents to USCIS after the case has been filed with USCIS.

If USCIS wants more information they will send a request for the additional information. Wait until you receive the RFE, then respond.

Did I made a mistake? If so, how can I correct it and would it actually make a difference in the processing?

You are required to wait one year after having asylum approved before you can apply for residency. This normally causes a rejection immediately. ...

You will need to hire a lawyer to assist in your situation. Present all of your documentation to an experienced immigration attorney.