My politicalasylum is still pending for the interview(4 years already). I want to see my parents in Puerto Rico. Is it possible?

Puerto Rico is within the US territory, and you are free to travel there.

How long till my interview?

your guess is as good as anybody but on average I guestimate it to be between 4-6 years at this time but you can check monthly with the DOS visa...

My girl friend and I are from Japan, she has I20 and I have a green card. Now I would like to marry her. What do I need to do?

you can go to the justice of piece court in your town or see the mayor and get married, but it will take 2-3 years before she can get her residency...

Notice of Action letter for my I-751 ..... asking for more avidance

you should write a letter explaining why you don't have what they are asking.

a serious non political crime will affect your asylum, but not this.

NVC Affidavit of Support and Visa Process

you can send the 864 and taxes etc now but I would wait until DS-260 is completed and then send all evidence at the same time to save on processing...

Nvc or uscis

yes you can and NVC will send you a letter shortly after USCIS notifies them of your 601 a filing halting their processing.

Am I cap exempt if a new H1B petition is filed (due to an approved/activated H1B from 2007/08)?

you guessed right as a H-1 is only has a 6 year shelf life.

I have overstayed my visa and currently in Atlanta, Planning to visit Puerto Rico, Is it safe to travel just on Passport??

not if you go through a check point since you are here unlawfully.

Planning to marry in US, will it be change of status(expired L1 to L2) or fresh L2?

you can not apply for visa from within the US, you can do a change of status but what do you mean expired L1-b?