Are we eligible for asylum?

Under the circumstances you should file for asylum. The two year residency requirement will be superseded by a grant of asylum if you qualify.

Can I apply H4 Visa while H1B is under processing? Does H1B has any impact?

No, there should be no impact on your H1B if you file H4 based on wife's petition.

Hard to say how long it will take. You can hire an attorney and have them communicate with the consulate.

After our interview, the immigration Officer asked us to produce my husband's late wife of 1975's death certificate which we

Your husband should try to get a certified copy of his wife's death certificate in the jurisdiction where she passed away.

I NEED ADVISE ON MY CASE, I'M RE MARRIED AGAIN & I RECEIVED A LETTER SAYS I SHOULD DEPART THE COUNTRY. I'M LOOKING FOR A LAWYER

You should begin the green card process again with your new wife, as soon as possible. You should also seriously consider hiring an attorney.

Asylum application fill 2013 asylum office send ,,,

Unfortunately, I can't make out whether you have a question.

LPR with child

As an LPR you can file an I-130 for your child but he will have to wait for his priority date to be current before he can apply for a visa, and...

How can I get a worker permit?

Have i-130 and I-485 been filed? If you entered legally and are married to a citizen, you are eligible for a green card and for work authorization....

MSC I290B TO AAUnit. But the website for missouri service centre doesnt show a time frame. Any idea

Hard to say. It will probably be several months.

J-1 to F-1 & TPS

I agree with Mr. Segal, whenever you think is best.