GC filing if I remain outside US for certain period of time.

IT is impossible to answer your questions without knowing on what grounds you are seeking to apply for green card and if you are in status now.

How Can I apply for H1 CAP in 2017 if my L1 A 6 year stay completes by June 2016

I do not believe that L1A category provides for any extensions under AC21. You need to be in H1b status for it. You should meet with an attorney in...

NATURALIZATION -N400 OFFENCES

It depends on actual certificate of disposition. If you admitted committing the offense during the plea, it equals conviction for Immigration...

Is there a contradiction between the concept of TPS and J-1 ? should I expect a problem with my plan ?

You have a complicated situation and you need to see an experienced Immigration attorney in your area. Different options yield different results,...

In this situation can I apply for AP ? can I leave country for 3 weeks ?

You can try filing for AP, however, if I 485 is denied , depending on your current status, you might have troubles coming back.

Are you dealing with a fiance visa? You need to see an attorney in your area.

H1 to H4 COS - Resign H1 job on COS online approval or till getting receipt?

You will have a solid proof of H4 once you receive an official notice of approval of your COS application by mail .

Can I extend one more semester graduate in may 2016 and apply opt to be safe? how long will it take F1 reinstatement currently ?

Every case is different and sometimes cases take longer than expected. You need to seek advice of an experienced Immigration counsel in your area.

What is the best way to apply for a green card or can we change into other non-immigrant status?

It does not appear you qualify for adjustment of status based on the facts provided. However, you should meet with an attorney to talk about it .

What is the best option for me, appeal the USCIS decision or submit my case to a immigration judge??

The EAD is invalid as the underlying status is denied. You need to meet with an attorney to talk about your case.