Do I need to file for permanent residency right after getting married?

You don't need to stay on F-1, just apply for green card and get your permit. If you would like to travel, you can get advance parole.

GC on EB1 category by another employer

Depends on the category.

You can do it. I guess it will be easier for you to travel there...However, remember, where there is duel citizenship, there might be duel taxation...

I am granted asylee from the judge when should I apply for the green card?

You can apply after a year of being in the asylee status. Check out the instructions to I-485 form on uscis website.

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Will reporting oversea income on my Tax Return negatively impact my application of Form I-751 (removal conditional PR status)?

I don't think they will think that since you have been there for 2 months only, provided its not a permanent position. As to how to report it,...

Can i seek asylum in US ?

You can, but you need to be in the US. Read instructions to form i589 application for asylum on www.uscis.gov to have an idea of what you need to do.

Can I still applay for the dream act if I have done voluntary exit?

You need to seek a consultation with an attorney. It's impossible to accurately answer a question like this on this forum.

Could my fiancé stay or file some sort of waiver?

Illegally returning after deportation may present a great problem, before making any conclusions, an attorney would have to study your fiance's...

K-1 Fiancée Visa

You will have to provide fiancé's address abroad.

The lawyers makes me confused

If the charges were dismissed, you can apply. Be ready to explain what happened.