Mystery snails (and I'll assume its the common species sold, bridgesii) mostly eat dead vegetation and/or algea. If niehter of these two sources are present though, it MAY eat your live plantation, although thats only if niether of the first two food sources exist. There is no great way of knowing if it will be able to co-exist with your betta other then trial and error. In most cases, the betta will try and nip at the snail, causing tentacle or partial skin damage. If this persists then no, your snail will not be able to live with the betta. However, this is solo dependant on the personality of your betta. I have had a gold mystery snail (and there is no diffirence in the color of a snail vs its needs/etc) in my bettas 10g for a few weeks, and other then the recent 'illness' my snaisl contracted, the betta doesn't touch it. Itdid tip at its tentacle once when I first introduced it to the take, but left it alone afterwards.
Snails do possess the abilty to regenerate lost tentacles and small skin portions, but to a lesser extent. For instance, a repaired tentacle will not be as long as it use to, and may stay stunted forever. Just be aware that snails, in no ware, are good 'cleaning crews' contrary to popular belief. They produce much more waste then they clean up, and due to a microorganism that lives in thier intestines to help digest food, they stoll will often cloud water.