If PPBUS Voltage is 12.56~ and you have 3.42V on SMC RESET unlikely. If PPBUS voltage is 12.23~ check SMC RESET voltage. You should have 3.4v. If you dont, the SMC may have crap under it or the Reset chip isnt working.

Usually, a battery issue will not cause a no power up issues. Batteries in MacBook Air’s rarely swell or short. When you plug it in, I want you to look at the charger light. Does it light up? Unplug it, and hold down power for 10 sec. Put your ear up to the vent and plug it in again. Do you hear the fan spin up?

Backlight failure other than the screen on these is rare. Fuse failures are almost non existent unless you plug the screen in with power. It is probably a LCD issue. Of course if you have obvious corrosion on the board, address that before blaming the LCD. I’d put my money on the LCD though.

Something got messed up in the audio circuit on the board. There is no sound card. The audio decoder is on the board. What concerns me is the restart. I doubt the CPU took a hit but it is always possible. I would pull the board and inspect the circuit under the microscope if you have one, If not, your best bet is with someone that does board level repair such as myself or Rossmann Repair Group.

EDIT: RAM!!!!! RAM will cause horizontal lines!! Try a new ram stick or try them one by one in different slots. DO NOT put it in the over like the previous answer suggests... DO NOT clean it with alcohol. It is 100% useless in this case. How long ago did you replace the board? These boards suck. You probably got another bad one. Unplug everything but the dc in board and the screen cable (Including battery) plug the charger in. Does it work? If so. narrow it down. Start plugging things in one by one and see what screws it up. If it still does it with everything unplugged. New board. Get a 820-3330. It will work and wont die. You may have to modify the LVDS connector a little but that's it.

Dud battery. Was it doing the same thing prior? If not, get this battery swapped. Dud batteries happen.

Your going to have to pull it from a donor board. Use lots of flux, and heat the donor board from the bottom from directly below the connector. You will need a beefy hot air station like the Quick 861DW. To solder it, your going to put some leaded solder on the pads, wick it away, add new solder and flux, place the connector then bottom heat again on the patient board. There is nothing easy about this. It is a real pain

I would try a new HDD flex cable before anything else.

So with these, pretty much any chips we can get for them will not last very long. Most supposed "New" AMD chipsets are either rejects or reballed garbage, so while soldering a new chip to the board is not a problem, getting a good chip that can last and be reliable is. Sellers on ebay that claim to replace the GPUs are more often than not heating the chip up with a heatgun. Heat will make the chip work for a short amount of time. Others will claim to reball the GPU. This again will only last for a short amount of time. The issue is not the solder balls. It's the bumps in the die itself. The heat involved in reballing will do the same thing that a $30 heatgun from home depot will do. These machines are nice as far as serviceability... You can replace the battery with easy contrary to the 2012 and up Retina. If you have a 15 inch, I'd say get a 2012 board for it. These use a different chip and will work. You may need to do a slight modification to the LCD cable to get it to fit in the new board. The board...

1. Rice does nothing. Literally nothing! 2. Your going to have some corrosion issues down the line on the board. It's inevitable, unless you clean things up. 3. Your keyboard issue is probably just that, a bad keyboard... Not impossible to replace the keyboard on this model. Yes, it's a pain but not impossible. The keyboard is held in by 50 or so screws and/or rivets that you will need to cut out and replace with screws -- Thank Apple Engineers for that wonderful idea -- 4. You will need to pull out the board and the I/O board to get to the keyboard so a good ultrasonic cleaning of the board would definitely be something to look into -- if that fails, a soft toothbrush and some 91% isopropyl alcohol should work fine. 5. Prolonging cleaning of the motherboard is just going to cause more problems. I'd pull the bottom cover off and unplug the battery and get that board cleaned ASAP. You got lucky.. It still turns on. Most Retina's we see here aren't so lucky and have board issues after a spill.