Sick Betta - Severe Fin Rot and Velvet (?)

Pics (potato quality, the lil bugger wouldn't stay still): https://imgur.com/a/K1DNp

Does he have velvet? Or is that ich? What do?!

Full story below~ Sorry for the novel, I'm just concerned for the fella, and I want to try my best to save him (and protect my other fishies)!

7/16: Won a betta from a church fair, added him to our 10G tank with several platies. (I know - shame on me for not QT first, but I just don't have the room! ><) This is my first time owning a betta, so I was concerned he'd harass our platies, but he's not aggressive at all - hooray!

Several days later, his tail is looking a bit ragged. None of the platies really bother him other than the occasional derpy one bumping into him, certainly not nipping. I thought he might be tail biting because of the current in the tank, so covered the filter intake with pantyhose and baffled the output with a water bottle. As additional precaution, I removed a couple plastic plants. He's doing OK otherwise.

7/24: Had to go out of town for a few days for work, so right before I left I fed them, did a large gravel vac/water change, dosed with a bit more than usual Prime, and then covered the tank with towels.

7/28: Came back to see that Mr. Pinchy the Ghost Shrimp had died sometime over the weekend. :( Tested w/ API and saw that Ammonia was at almost 1.0, eek! Did another vac/PWC and dosed with more Prime. Tested again and it was down to 0.25-0.5.

At this point the betta was looking rough. He had lost most of his fins and he looked like he had spots all over him. I assumed it's Ick.

7/29: Went out and got API Super Ick Cure and aquarium salt. I have a couple fry in a breeder net, so I wasn't very comfortable with dosing meds but I got it just in case the heat/salt method failed.

Over the course of the day, I got the temp to 86F and aquarium salt up to 1 tsp/gal. Added a bubble wand to help with oxygen levels.

7/30: Noticed the platies were acting fairly subdued and were gasping at the surface a lot. The betta had been flashing a lot and now has stopped eating and occasionally getting clamped fins. I started to question whether it is ich or velvet, since the spots look more brownish than white on his tail.

Since the platies didn't seem to be handling the temps well, and the betta was looking progressively worse and I wasn't even certain what I was dealing with anymore, I decided to medicate him separately. I cut the top off of a gallon water jug, dosed with a proportionate amount of API Super Ick (malachite green, so it should be effective against both ick and velvet right?) and am now floating it in the main tank to keep it warm @ about 82-83F. Covered everything with towels again in case it is velvet to help block photosynthesis. Ammonia was about 0.5.

7/31: Took a peek earlier today. Betta looks a tiny bit better, fins aren't as clamped and /appears/ to have a bit less spots on his tail. (Hard to tell though, with only a top-down view.)

So my questions are:

1. Is this velvet? Ich? Both? Or something else entirely? His pectoral fins look like they have white spots on them, but the spots on his tail are definitely more tan/brownish.

2. With the meds, should I refrain from changing the water in his jug? The instructions say dose again in 48 hours, then after another 48 hours perform 25% water change and add carbon to filter. I'm concerned for the ammonia levels and fin rot, but I don't want to lose medication effectiveness for ich/velvet either.

3. I got a 20H tank I've been holding off on moving them to. Should I refrain until everything is clear, or would it be OK to net my platies into the new tank? I assume everything in the 10g is infected (gravel, decor, etc.), but the platies show no visible signs of sickness. Would moving them now just infect the new tank?

I was thinking of moving the fry and platies to the 20H so I could dose the entirety of the 10g (and betta) with malachite green without having to worry. But I don't want to introduce it to the new tank either (in case they have it hiding in their gills or something). If I move just the fry and their basket, would that still be enough to infect the new tank?

Thanks for your help!

TL;DR: What is my betta sick with, and what would be the best way to treat it? What should I do about other tank inhabitants (platy)? How infectious is this?