Can we resign and serve notice period in US if L1-Extension is in process or L1-Extension has been rejected?

You can resign from employment at any time, which will likely result in a withdrawal and/or denial of the application.

Does the 6 year quota starts from H1b approval date on I-797 or the actual H1B start date ?

The 6-year limitation is based on time actually spent in H-1B status.

H1 b visa- my colleague at work got her visa transferred from a body shop company which had hired her on an hourly rate .

These are very suspicious facts. She should consult with an attorney.

I file i130 for my wife that is in Ethiopia this pass April 6 and its pending, we did the visiting visa but rejected cause the

Once the petition is approved, the USCIS will forward the case to the NVC.

Green card denied due to lack of requested evidence but motion for appeal accepted & now in process. Can I renew my work permit?

You can only renew a work permit if the I-485 application is considered "pending."

Overstayed J1 visa for 3yrs now... ways to get back in status aside marriage

If you are out of status, you have limited options available to you, if any. Consult with an attorney.

No, you cannot transfer an affidavit of support to benefit a third party.

Work permit

Assuming you are eligible and properly filed the application, it could take 2-3 months.

Asylum applicant work authorization. How long does it take, can it be expedited, what are the benefits after the 150 days?

You can apply for the work permit as early as 150 days after the I-589 was filed. It can take 2-3 months for the work permit to be issued.

How can I create a company being an International student under an F1 visa?

It may be possible to "start a company," however you must be wary of violations of status. Consult with an attorney.