Agree on all points.Those are common pond snails. They are avoided like the plague by many aquarists. I personally keep them as part of my competing snail populations (I have 5 different species in a tank). And I also have no real problems with them eating my plants, the exception being when they munch on weak leaves. This is a yin-yang thing though, as while I hate having holes munched in the leaves, it is also a very clear, visible reminder that something isn't right in my aquarium. Usually it's an indication that I didn't dose ferts the previous week.
If you don't want lots of them, kill them all now. Without any other competition in the tank they will multiply quite readily.