Get a length of pantyhose and run it along all the decorations. If any of them cause tears or runs in the pantyhose, those will be the culprit for the fin damage, too. Remove any problem decor, of course.
Some very powerful filters can catch a Betta's long fins in the filter intake, and he can tear them escaping from it. You might consider adding a pre-filter foam to rule out this possibility.
If the edges of his fin are dark, red, or slimy he could have a bacterial infection called fin rot that will cause his fins to deteriorate and tear. You would normally treat with an antibiotic for this. Since Melafix is a mild antibiotic you are probably already covered if this is the case.
In a small number of cases, there are Bettas that will intentionally bite off pieces of their own fins. This can happen from stress or boredom from a variety of factors. I had one that did this - I finally found out the light was too bright for him, and a dimmer light fixed the problem. But other people have found that their Betta does it from too strong a current, improper temperature, too small of a tank, not enough variety in tank decorations, etc. The damage from tail biting is rather unique looking, they'll be missing large chunks deep down into the fin while the rest looks fairly long and even. This is much less likely to be an issue than the others, though. Watching him carefully and extensively, you should be able to catch him in the act if that's what's happening.
On a side note, it takes 4-6 weeks with a fish or other source of waste in the aquarium and a filter running to cycle an aquarium. With a fish already in the tank, you are doing a fish-in cycle. Daily water changes to combat the spikes in Ammonia and Nitrites that will inevitably take place during the cycling period would be a good idea. Especially if he has fin damage. Even if he doesn't have fin rot yet, it can easily set in as a secondary infection if the fin has already been damaged, and that's especially likely in an uncycled aquarium. So keep the water pristine with daily water changes to make sure he doesn't get sick. After the 4-6 weeks are up, you can go down to weekly partial water changes.
Oh, and I don't recommend Apple Snails with Bettas. That's because the long, thick antennae of the Apple Snails can prove irritable to the Bettas, and they are known to bite them off. That will be quite painful to the Apple Snails. Instead, you would be better off getting a snail with clear antennae that don't induce a feeding response in the Betta. Personally, I've kept Olive Nerite Snails and Tracked Nerite snails with my Bettas, both of which have hardly noticeable antennae, and these have worked very nicely. These are also better algae eaters than Apple Snails, and can not breed in fresh water. Nerites would make a much safer alternative to Mystery or Apple Snails. Don't forget to add in a calcium supplement, such as a small piece of unflavored cuttlebone, for your snails.