Yes you need you send i-765 which is EAD application with it and i-131 for travel documents (i-131 is optional if you need to travel, however, no body is recommending to travel lately).
You should send i-485 if you want EAD faster. If you thinking to send just i-360 to get faster EAD than right now process time is 48 months which is 4 years.
Yes depends on where he/she lives. If it is for you and you reside in NY then that is the correct address.
Follow this link based on your resident state.
https://www.uscis.gov/Certain-VAWA-T-U-Filing-Locations
Usually it takes 30 to 90 days. Like Rhema1 said it might has to do with your country of origin since it is one of the ban countries. How was your interview? How would you rate officer's behavior towards you? Did he say anything at the end of your interview?
Yes USCIS accept the USPS mailbox however you still need to put your physical address where you reside. Answer to you third question is you can put YES so when your VAWA approves they will send you C31 EAD meantime you can send C09.
Congratulations. It is too early to reach out to your congress person as per my experience. I would contact them after 6 months and that's what i did after my VAWA approval. Other than that you will be fine just enjoy the moment. It is a great feeling having VAWA approved after long wait.