What is my current status while my I539 application is in process? Is it still L1? May I get back to work immediately?

If there has been no adjudication on your I-539 then you may be able to withdraw it and continue in L-1B status. Further review is required.

Can my employer file H1b when only on EAD of TPS but TPS application still pending?

You may very well be able to file for the H-1B if you were not out of status for more than 180 days but you wont be able to do a change of status...

I140 RFE responded but still status shows 'RFE mailed'

The status link on the USCIS is often incorrect or slow in updating the status on a particular case. I would not rely on it.

Not at all. I have had many marriage cases where the parties met on a dating website.

Can I continue to drive beyond 02/05/2016, while I wait for decision on my visa extension?

The NYS license is valid until 7/8/2018.

It can be done in the USA. The case needs to be analyzed for pre-conceived intent. Many people have come to the USA with a visitor's visa and...

Can i apply for NIW- I140 petition while on J1 visa, pending J1 waiver approval?

You can begin working on the NIW case but you would need to get the Conrad waiver before completing the NIW case.

First one needs to analyze as a TPS foreign national, whether you qualify for an H-1b visa filing.

Can she apply for adjustment of status here, while her change of status application is still pending? Thank you

We would need to know when her spouse can apply for naturalization? That may very well be the best option.

If I received a check from the Government. It could affect my eligibility for a green card or citizenship

This will not affect your immigration eligibility at all.