Yeah looks like a pond/bladder snail.
As long as you dont over-feed, you wont be over run with the snails though. They wont reproduce unless there is plenty of food available for them.
I've only ever seen like 4-5 in my 26 gallon that they happened to hitchhike into.
As long as your PH is above 7 and you provide some calcium supplement once in a while you should be okay.
AS for shrimp. I'd suggest getting ghost shrimp first just in case your betta eats them, that way you dont lose too much. I've got ghosties in 2 of my tanks and none have been eaten, they can get away pretty fast, but for the first few days the betta's do stalk them thinking about having a taste to see what they are.