It may be considered compensation and thus employment. Wait until you have employment authorization before beginning employment.

I am on grace period. What should I do next?

There are two obvious choices based on the limited information you provided. The first is to seek change to a B-1/2 or second to return the school...

For I130 petition , does USCIS send the RFE after receipt of the package by intake staff or after docs checking by adjudicators

The receipt is sent when the correct filing fee is submitted and there is initially on the face of the submitted documents a claim of eligibility...

Yes it is possible to return the United States as a lawful permanent resident after having been removed from the United States. He will need to...

Does USCIS contact petitioners by phone or just per regular mail and do we have to inform about mobile number change?

USCIS contacts are normally by mail. If the I-130 is under investigation or field review phone contact may be made.

If i have an I130 petition pending ?

First you receive the receipt notice. Then it is possible to receive an RFE if not then the interview is scheduled and at the interview questions...

I have an I130 application pending with USCIS, if i have a problem with application can attorney intervene to resolve this prob?

Need to know what the issue is and the attorney should be on the record with USCIS

You should discuss this with your attorney and have attorney follow up with Immigration. And yes Immigration makes mistakes.

My wife wants to file a petition for my mother in law to move in but do I need request an extension of her stay first?

Yes you should apply for permanent residence based on an immigrant immediate relative petition and adjustment of status application. There is no...

Recently become a permanent resident, do I qualify for the stimulus check?

Yes, you qualify and it does not impact your immigration status nor any future applications for US citizenship.