Is it possible or how to apply for a second part-time job H1B petition?

Yes it is possible.

My R1 case is in the appeal process, which was denied, can I be put on payroll if I was not before, during during this process?

You cannot work since the R1 was denied, unless you have employment authorization based on some different application.

L2 I-94 extension.

I agree with my colleagues

I agree with my colleagues. You may want to read the instructions and the form itself as well.

Separation after applying for the 1-130

You need an attorney. If there has been domestic violence in the marriage you may have other options available to you.

Do I have to work with same employer after I my Green Card is approved ???

I agree with Atty Behar

Dear Sirs, Please advise me how to follow and check my I589 application.

I agree you can have your attorney contact the asylum office but I think it has not been too long since you filed. Since you received the...

Immigration VAWA - is there any exceptions to requirement that the abused spouse file within 2 years of divorce from the abuser?

Try to make the effort to consult with an attorney. Only after an in depth discussion with you can an attorney make an informed determination and...

When would I become eligible to apply for naturalization based on the information given here.

I agree with atty Behar

Is it best to file i 130 now and next the waiver once i 130 is approved or to wait and see if new law passes in june. question

That is too difficult a question to answer without more information. Consult with an attorney for a detailed analysis of your situation.