As long as you are not getting paid, it should be okay to publish.

Why this is no movement for Aug 2013 for the priority date for EB2 and EB3

I agrer with my colleague. There are simply too many applicants.

Can I work legally part-time somewhere else while on full-time H-1B position?

You may as long as your part time employer files a H1-b.

You should be fine as long as yiu stay in us territories.

What papers I can send my mom in different country when she aply for B2 visa?

I agree with my colleague. A letter of invitation and an I-134.

How do I change to my married name on my employment authorization card?

Best to consult an attorney. You can file for a new EAD based on your marriage.

I want to cancel my approved religious asylum case, my home country became safe to live in, what are the steps to follow?

Consult an immigration attorney. I do not recommend cancelling your asylum.

Do you still have your old passport with your maiden name? If so you can present that when you travel. Generally, it takes 90 days from filing...

I was approved for an F-1 visa, accidentally booked plane ticket, to arrive 31 days earlier

It depends on where you enter. Probably best to rebook your ticket.

F2a current aug 2013

Need more facts. Is your spouse a us citizen? Consult an attorney.