Legal Question in Immigration Law in California

Re: Applying for I-131 while awaiting adjustment of status

"Is it possible to process the I-131 in advance?" No.

"Do we need to be married before application?"

You need to have applied for adjustment of status in order to be eligible for advance parole (and you need to be married to apply for AOS).

"Do I need to be in the US for application?"

Yes, and for the approval of the application -- the latter is the problem here.

"And if so, can we get one in that small amount of time?"

Not through the usual procedure. Whether you could get AP from the local USCIS District Office would be up to that office's discretion -- they can, if they want, issue AP on an emergent basis. Because your AOS application would not be on file with the local office, however, it might be hard convincing them that you are eligible for AP.

Without knowing more about your case, I can't give you any advice, but two suggestions that might be appropriate are: (a) marry in a civil ceremony immediately (or as soon as the K-1 visa is issued), them immediately apply for AOS and AP, asking for expedited treatment of the AP application; or, if even that won't allow for enough time to get the AP before you have to depart, (b) try to make advance arrangments with the District Office (though this will be very challenging; probably impossible in most places).

Good luck!

Re: Applying for I-131 while awaiting adjustment of status

You need to get married and file I-485 and I-131 immediately and request expedite process of your I-131 application. You have to get the I-131 approved before you leave US. Otherwise the I-485 will be deemed abandoned.