H4 extension ?

They don’t need an approved petition nor a valid visa. Their stay in the US is controlled solely by the I-94.

You need to consult with a Board Certified immigration attorney about the possibility of deportation before you worry about naturalization.

Probably not but the first issue is whether you are subject to the two year foreign residency requirement.

Yes, they should file I-485 and I-130 together along with much more evidence and and many more forms.

What will happen to my application for extension of i20.

The International office at your school will take care of that.

Is there a chance to convince the U.S. Department of State to recommend a waiver for 2 years Home residency requirement?

You are correct that the chances of getting a waiver for US government funding are almost zero.

What should I do AOS or consular process?

Under these facts, you should wait for adjustment of status and file in the month your PD becomes current. You will get travel and work permits...

Social security after filing for adjustment of status ?

About six months but it varies. Probably 4-6 months is the range.

Question about US citizenship through marriage ?

One of the rules is that you must be "living in a marital union" for three years. That implies marriage but it also involves issues regarding the...

I have two questions regarding i175 condition greencard

The receipt notice you received when you filed the I-751 has the extension of your conditional status. See how long CIS gave you on the receipt. ...