I am still pending green card through political asylum

You haven't mentioned the one thing that could help you; get an attorney.

The most unanswered question in the law of US non-immigration visas, Which organization is in charge of 306 section refusals?

You don't seem to have a legitimate immigration question, other than a general request for help. Perhaps you have already done this but you should...

I'm a LPR and lived in USA for just about 2-3 weeks per year(studying abroad) would i be able to sponsor my husband(co-sponsor)

Aside from questions about your long absences from the US, and how they may affect your permanent residence, which you may be able to explain with...

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I really think that the main thing that you need to do now is get an excellent immigration attorney to help you on this waiver.

No they don't, but you probably should let them know..

730 just granted leave USA without a travel document.

You should send her the travel document once issued.

Can I move to another State while I am waiting for a decision on my pending (2 years) Asylum case?

Yes, you may move. Just remember that you have the obligation to inform new address by filing form AR-11.

it will soon be cashed, don't worry.

Petition thru marriage with us citizen ,spouse entered unlawfully from Mexico

You need to consult an attorney with more detailed information but it appears that she has a good chance of having her provisional waiver approved.

Can we make petition for fiancee Visa from out side USA

yes you can, as long as you plan to come back yourself to reside in the US, an consular officer believes you.