Depends on the broker but it’s usually LIBOR plus a mark-up. The mark-up depends on how much you borrow; the more you borrow, the lower the mark-up interest rate.
You can leave a short in place as long as you can meet margin requirements(if the trade goes sideways; price of stock increases). But margin rates are so high, it can be as much as 8%(LIBOR + mark-up) or more, that it is cost prohibitive to short a stock for a very long period of time.