Am I subjected by 212(e) 2 year rule?

Probably not as most Au Pair programs are not subject to HHR.

First time filing OPT application (I-765 form) after graduating

I agree about the need to have a professional consultations but I think you will be fine.

First time filing for OPT (I-765 form)?

You can apply for OPT. No need to be stressed.

Should I file form I-90 if I lost my green card but the card is already expired expired?

No. this form is not applicable to you. You will have to wait until your remove conditions before you would get a new LPR card. You can rely on the...

I was given at week but it was not written in my passport and i stayed 1 month and 3 weeks. will my visa be cancelled?

If you violate your status your visa is deemed invalid.

I had Medicaid is it affect my citizenship application?

Depends on how you got the LPR status to begin with.

When is a fit time for filing I-765 form (OPT) ?

You have a window of 60 days before through 60 days after the graduation to file. Talk to the DSO.

Can you sue a immigration lawyer in Raleigh NC

"Can you sue a immigration lawyer in Raleigh NC" is not an immigration law question but one under NC law.

I agree. They are probably having some issues with her visa application and would like to ask her questions under oath.

What should I do as I filed incomplete documents with I-765 EAD?

If you got a receipt for I 765, they will ask you for pictures on the RFE if they need those.