Can I file the AOS paperwork?

Attorneys are not goods:) you should have a good working relationship with your lawyer. Call a few. Get few quotes and go with the one you feel you...

Traveling out of the United States - issues with naturalization and 3 months residency requirement for naturalization

You would benefit from a private consult with a lawyer. This is a general, public forum.

How long do I need to wait more and what I would need to do?

Hard to advise without more facts and a private consultation

You can read the instructions here: https://www.uscis.gov/sites/default/files/files/form/i-751instr.pdf

When a husband and wife did not convince an immigration officer that their marriage was real at their first green card interview they usually get a...

New policy of USCIS for processing asylum application?

Yes of course. Thats why it is posted. Interviews as soon as 30 days.

Can pending asylum case affect your brother F1 visa?

Depends on things that an attoney would illicit and find out. Consult in private with an attorney, your brother should.

Will I have issues with CBP when entering the country?

It is ‘his’ application for naturalization and should not have any bearing on your reentry. To be safe, when traveling, carry all your docs with...

Will I get a visitor visa if my sister and her husband have applied asylum in USA?

You certainly can try. The burden is on you to rebut the presumption, so present the best evidence of your ties back home: a job, bank accounts...

You have a lawful entry but also unlawful presence. Only marriage to a USC will help.