Looks like plain old pond snails and they are a real problem, a nuisance and a big mess. You will also have a very difficult time getting rid of them, if fact they will almost certainly overpopulate unless you take drastic action very soon and continue it religiously.

My Petsmart keeps Their mystery snails with their aquatic plant setup and I've seen other stores in the U. S which have nerite snails in with them too. Though, I don't think it is any of those. I can't really see the photos too well as my phone likes to compress images on this site. But it's highly likely it's some sort of pest snail.

To better deter snails from making there way into tanks, you should run newly purchased plants under the tap and inspect them for any eggs snail or other hitchhikers. This also allows you to remove dead or dying leaves and etc.

Do you like the snails there?

You might soon have a problem with them if they are a live bearing species such as a ramshorn, but your tank being 1 gallon it shouldnt be that difficult to control if the time comes.

They shoukd be OK as snails are very resilient but I can't say for sure as I've never done this.

You'll have to wait and see and decide If they're really worth keeping. If you like the idea if a snail in your tank get one of the commercially available ones.

Mystery snails are always fun and come in yellow white and black. Though 1 gallon seems a bit small for the snail and for a betta.

The water conditioner shouldn't be in anyway toxic to snails unless it has a disclaimer about usage with invertebrates. Or has some type of copper or what not in it.

Well after carefully removing an egg sac and still having tones of snails in one week, we have switched to beta fish. See my new post. I think I got in overt head and hope to raise the beta correctly.