What can be done to expedite the process of getting EAD card for the derivative asylee?

Request that USCIS expedite the processing.

If I apply for US student visa ,Will it be rejected on the ground that my mother's form I 130 has been filed by my grandmother?

No it should not be rejected because of that filing by your grandmother.

Can my son join me and go back to study till graduate.

It is possible for him to receive asylum and then leave the US for some time. Hire an attorney to help.

How can I stop to deport her and stay here

The best approach to stopping a deportation is to hire an immigration attorney immediately.

Would it be better if he didn't file for I140 & I485 now and wait until H4 for his wife (fiancee now) is approved?

He should discuss this with the company attorney and his fiancé.

HOW OR WHAT DO I NEED TO DO TO GET FIANCE TO U S AND HIS DAUGHTER SHE IS 5 yrs old

You must travel to meet him at least once before you can file a K-1 visa for him. You can file a K-2 visa for the child.

All good questions. File the I-751 with the name you now use along with the documents showing the legal name change.

What am i looking at? This going to hurt my immigration status of daca.

Oh yes this can certainly hurt your immigration DACA status. It can even cause your deportation. Time to hire two lawyers. One for immigration...

I agree with my colleagues. You need to file an amendment to your application.

Yes write just one cover letter form the whole packet.