great news!
I went crazy in terms of months of searching and failures trying everything I could find.
I ended up going to my last resort... tearing into the laptop.
I used an iFixit kit and opened up the laptop and found my USB on the right side shorting out and causing the 170-degree hot spot on the right side. This 170-degree temp was seen using an app called TG Pro (free for a short time).
The "Thunderbolt proximity" temp (which is right next to the USB in suspect) was 160-170 when awake. While sleeping, the temp was noticeably hot to the touch. Even after sleeping for a short or long time, the temp still registered 160-170 degrees. No logs showed waking issues so I was desperate as I lost over 10% an hour!
I decided to open it up and see what's going on....
I used the how-to:
https://www.ifixit.com/Guide/MacBook+Air+13-Inch+Mid+2013+Logic+Board+Replacement/15388
It was a lot of tedious work, pretty time consuming (2h or so) and VERY RISKY for what I was about to do.
I ended up removing the right side USB port from the logic board by gently rocking the port back and forth until all the solder points broke away from the board. I removed all remaining pins using a pair of needle nose. I then applied a bit of electrical tape to cover the bare/exposed solder points left behind.
After putting it all back together (VERY CAREFULLY), I found that the temperature at the Thunderbolt proximity sensor dropped CONSIDERABLY. It now sits at a very comfortable 80-85 degrees. I charged the battery all the way up and did some normal work. All was good. The laptop functioned as it should (WIN! I put everything back together correctly! haha) I closed the lid (which was a normal battery death sentence after several hours) and left it overnight... To my great joy the next morning, the battery only dropped 1%!!!!!! YAY!
TLDR: If you have a hot stop on the right side of your mid-2013 air, and it's verified by an abnormal temp via the Thunderbolt proximity sensor, open up your Air and see what's going on. For me, it was a busted USB port that, IMO, was shorting out on the board, draining the battery and creating the heat. Surgically removing the battery draining cancer fixed my problem!
Images:
1. board and USB suspected to be the issue
2. board without the USB port
3. board without the port v2
4. the usb port that was removed
p.s. I am in no way a how-to writer, pro or anything like that when it comes to this stuff, this was the result of desperation and many, many weeks of trial and error. Please forgive the novice writing and especially the lack of documentation and pictures. I didn't have much hope when tearing into this Air!
Good luck and don't give up!