The damaged pectoral fin may or may not grow back, depending on where the break occured and why it occured. If the "root" of the fin is not damaged, it can grow back. However, it might never be straight again.
Pectoral fins in some species of anabantoids are actually rudimentary limbs (such as the Climbing Bass, which can actually climb trees and survive two days out of water). But in most species of betta and gourami they've become nothing more than display features to attract a mate - kind of like a peacock's tail.
Unfortunately, they can't feel things with them - they aren't "feelers". Neither can they taste with them or hear with them.
Just by observation, I think they do use them as feelers or at least as a guidance of some sort. When they get close to another fish, they will send those "antenna things" out, so they are using them for something. Whether or not they can actually have a touch sensation with them, I do not know, but I do know they use them when encountering objects and other things. So, they do use these for some purpose. Maybe to them, they are just a little whacking stick attached to their body. I have no clue.