I am on OPT visa applying for family based AOS. Should I include Advanced Parole if I don't plan to travel anyway?

Advanced Parole requires humanitarian, educational and employment travel reasons.

A Refugee Travel Document (RTD) can be issued to a lawful permanent resident who obtained such status as a refugee or asylee in the United States....

We are about to file my spouse I 130 application and I 485 application concurrently but we have some few questions in mind.

Yes, you must be truthful when completing the immigration forms. As to filing tax, you can file jointly or married filing separately, but as to...

What can i do ?

You have much at stake here, and I suggest you contact an immigration attorney for an in-depth consultation.

Can I travel on my old advance parole document?

No, a valid unexpired advanced parole is necessary for travel.

Will I face a problem in entry at JFK airport for ESTA responses?

Your question requires an in-depth analysis, I suggest you consult with an immigration attorney before you travel.

Can I travel to Mexico as my 10 year green card is expiring in 6/302018 ?

Yes, you can travel, assuming there are no other underlying issues that you did not mention in your question ... You may also renew your green...

In some instances criminal proceedings may be initiated as well.

Can a asylum applicant go back his home countary?

Generally, the refugee/asylee travel document is not available to asylum applicants.. however, even assuming you get one, travel back to your home...