Sick Betta - Severe Fin Rot and Velvet (?) (xpost from r/aquariums)

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 wall-of-text story here: https://www.reddit.com/r/Aquariums/comments/3fch0p/sick_betta_severe_fin_rot_and_velvet/

Ammonia spiked to 1ppm due to dead ghost shrimp while I was away, now lowered to about 0.25-0.50. Treating with Prime.

Tried a salt/heat treatment for the ick/velvet/whatever, platies didn't do so well. Lowered temps to about 83F, isolated betta and dosed with malachite green (API Super Ick Cure).

So my questions are:

1. What is this? Velvet? Ick? 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).

What 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?