Michilcha

1. You should be able to work now but also should apply for a re-entry permit because Sep is more than 6 months.  Or take and spend your PTO in the US before the 6 months.
 
2.  When your AP expires you may be able to extend it if you have not received your interview/GC.
 
3.  No, working for an aid organization overseas will not be given special consideration.

You can stay out of the U.S. until your I-512 (AP document) expires.
 
If I-485 is approved, you should immediately return to the U.S. You can board a flight to U.S. with your I-512. You must inform the CBP officer that you have an approved I-485. Do not expect CBP to let you board without your green card if you are going through pre-clearance in Ireland or Abu Dhabi. 
 
Once you have your green card in hand, you can leave the U.S. for up to 180 consecutive days, but you are also expected to spend more days in the U.S. than outside. So you would apply for a re-entry permit from inside the U.S.  You cannot apply for a re-entry permit from outside the U.S.  Your re-entry permit application might require biometrics. You will need to be in U.S. for that. You can leave the USA for up to 1 year and 364 days  (dating from when you apply for a re-entry permit) even if your permit is not approved and issued. In the unlikely event your permit is issued before your stay abroad completes, you should return to the U.S. to get it.

About 6 to 8 months.
It varies per FO. Some offices take more than a year.
I-864 is filled-out by the US citizen, and the rest of the forms are filled-out by the immigrant. In general, send the following three packets (*note that I-765 and I-131 packets are optional, but highly recommended) in the same envelope to the Chicago Lockbox:
I-485 with I-130 receipt notice, I-864 and supporting documents, electronic I-94 copy, filing fees, etc.; I-765 with electronic I-94 copy, etc.; and I-131 with electronic I-94 copy, etc. You'll separate each packet with rubber bands or fasteners: https://www.uscis.gov/forms/forms-information/form-filing-tips
On the I-485 packet, I-485 payment method (check, money order, or Form G-1450) must be the very first item of this packet.
 
You'll eventually have to go to a doctor that has Civil Surgeon designation from USCIS. https://www.uscis.gov/policy-manual/volume-8-part-b-chapter-4
Timing of the Submission of the Medical Examination Report
Applicants may submit the Form I-693 medical examination report to USCIS:
Concurrently with the immigration benefit application; or
At any time after filing the immigration benefit application but before USCIS finalizes adjudication of that application. If not submitted simultaneously with the immigration benefit application, applicants may bring the medical examination report to an interview or wait until USCIS issues an RFE requesting the medical examination report.
 
Here is the I-693 courtesy letter: "DO NOT MAIL DOCUMENTS AT THIS TIME."

No not at all and I am not exactly sure why mine had been moved (I can think of  bunch of reasons like location of move and the fact i have an expired green card). Yours may not be in Potomac either - its worth checking online with the live agent. 
 
Btw you asked some questions on another thread and went through the same process as you beforehand so let me know if you need any answers to your qs!