I have applied TPS(only tps,no AP,EAD) while my i485 pending(EB2). I have EAD+AP from i485 pending. Now TPS got approved.

What is the basis of your I-485? Family? Employment?

Question on attaining EDA for sibling based GC petition (i 130 + i 485 + EDA + Advance parole)

It seems like E-3 is going to be the best option for you.

In our Divorce decree there is no clause for me getting consent for travelling with the child, can I travel without a consent?

Does your child already have a passport?

My sister is on a H4 visa and she is applying for admission for her masters. Can she be a sponsor for herself?

Nothing needs to be notarized, but she can simply study on H-4 if she wants.

My H1 extension is pending. If I apply for H4 EAD before H1 arrives, what status will I be in when these approvals arrive ?

If your change of status is approved AFTER your H-1B extension then you will be in H-4 status.

F1 status Reinstatement

Send them everything they ask for and it may be approved.

Been laid off a day before my planned vacation abroad (on L1A). Will I be allowed to enter the country when I'm back in a week?

You may be but if you are specifically asked if you are still employed you will have to honestly say no which may result in being prohibited from...

Cap exempt or not? What is the term eligiblity if cap exempt?

Recently USCIS has been taking the position that people who did not enter the U.S. in H-1B status have NOT been counted towards the cap.

What will happen to me when I will arrive to usa with a (SB-1) immigrant visa refused? what will be my status if I enter in usa?

You will most likely be denied entry. If you aren't it is up to the CBP officer to designate your status as a returning resident.

have you approached the violators about this?