Rushing my parents paper (using expedite based on urgent humanitarian - Immigration)

The process you have started takes an average of 5 to 6 months. Be patient.

I need some advice for a friend who is illegally here and wants to become a citizen. What is the best way to do that?

His best option is to talk with an immigration attorney.

Your current status is Temporary Protected Status.

Can I use old priority date

You can use the old priority date to establish eligibility for 245(i).

Will I face any problem with it? Will my green card enough as identification? What is your advice?

You should also take your passport.

If ordered deported by the IJ can the person be eligible for EAD under the final order of deportation section on the form?

No because it is not yet a final order. Once the Board of Immigration Appeals make a decision on the appeal it is final.

A key immigration law is pending for illegal immigrants...!!!

There has been no change in the law yet. Bills are being considered by Congress.

Yes you can apply.

Visa status change

The I-94 determines the end of his status. You should be able to change status.

Do you have to file I-130 etc..before filing a I-601 waiver, is it worth all the time and money if you can't show extreme hs?

There may be another route but you haven't given enough information. Consult with an experienced immigration attorney.