Agreed, I'd say Chinese Softshell. Good job rescuing it, turtles must taste pretty great because they're ridiculously popular over in Asia. From the sound of it that infection was probably just caused by bad living conditions, you'll have to see whether or not they continue to clear up.
I've personally never kept softshells, but I do know that they will go after basically any fish... I've learned the hard way that mixing turtles with fish isn't really as cool as it sounds. The only things I keep my turtles with are natives, feeders, and other turtles, and I always assume that eventually the fish will get eaten. So yeah, definitely give him his own tank.
I also know that softshells tend to be pretty nasty... I doubt Chinese softies could be worse than Florida softshells or spiny softshells, so you'll have to find out for yourself (some turtles, like snappers, can be tamed to a certain extent, but it really just depends on the turtle). As long as you don't have to handle him a lot it's not really a big deal.
It's up to you if you want to give them a basking area. Right now it would probably be ideal because he/she will want to dry out to get rid of some of the bacteria, it will also help it shed some skin. However, for species that very rarely bask, I usually just give them a flat rock or something for them to sit on just below the surface, so they're still submerged but can at least stretch their neck up and get some heat if they want.
As for feeding, well, turtles will normally take whatever you give them after a while. Shrimp, scallops, earthworms, tilapia, bloodworms... The list goes on and on. If you want to go pellets, I'd recommend Mazuri, Omega One, or Reptomin. IIRC softshells are mainly carnivorous, but you'd probably want to try and mix some veggies in there as well. As for live foods, I wouldn't even bother with goldfish or rosies, guppies are good if you can get them but wild caught minnows and CRUSTACEANS are great. Snails are also a nice treat, some turtles take to them, some don't.
Make sure to provide UVB lighting & vitamin supplements. I used to just use calcium blocks, but I learned the hard way that isn't really enough. Keep us posted on how the turtle is doing!