Epistylis is not as common as it's made out to be by certain people online. It is an opportunistic parasite, unlike ich. Epistylis lives freely in the environment, with no host. It only attaches to other animals if that animal is weakened by a primary bacterial or fungal infection, especially if that infection is external. Ich, on the other hand, is an obligate parasite. It cannot exist without a host, so the fish can be perfectly healthy otherwise and still get ich.
I can tell you with near-certainty that what your fish have is ich. Personally, I would not bother using the medicated food unless the ich meds don't work for some reason.
I doubt the wood had anything to do with your bettas fins. If it was a very rough or sharp piece of wood, it might tear long fins. However, your betta's fins do not look torn. They look melted away by fin rot. It's extremely common in pet store "rescue" bettas. If you bought the betta looking that way, then the fin rot is not your fault by any means, and I'm glad you've seen some minor improvement, but it is now your responsibility to properly address it and treat it. That's the danger with "rescuing" bettas from pet stores. It feels nice to give them good homes, but they often come riddled with diseases which may or may not be treatable. It's always a gamble.
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