I am having H1B visa and been working in US for 1.5 years now. Am i eligible for the jobs that require public trust clearance?

Attorney David has given excellent advice.

Can I overstay my status expiration whilst waiting for my I-360 approval? I know about the 180 day, I want 2nd opinion of risk.

You would need to be in status in order to file your I-485 Application.

What will be the next step from nvc once they get my police letter from Qatar? will they give me the interview date?

Police certificates from all countries will be requested.

Should I call USCIS to open an SR?

Take Attorney Jeffrey's advice. Good luck!

Postpone Expedited Immigration Hearing

You can try but as the other attorneys note it may be difficult

This seems very strange. What happens when you enter your receipt number into the USCIS website?

1. Even though I have switched my employer, does it mean that my H1B stamped visa(with my previous employer) is still valid?

A person can use a visa with a former employer, but will need to bring evidence of the NEW employment during travel (H-1B approval, paystubs, etc.)

Our big concern is, with the expired green card, if we have her sanctioned will she be deported?

What do you mean by having your mother in law sanctioned?

Can i sponsor my trip for visitor visa on the basis of my over seas bank accounts or funds? Who could be best sponsor?

Your question is a little unclear. Are you saying you have been working in the U.S. since 2009?

How can I stay few more moths in USA?

If you have the two year residency requirement then it will be difficult for you to stay/change status.