An H-bridge can be used, but also a relay.
Not if you need to be able to make one side both hot and cold.
I have used many pelter elements and all have been reversible by switching the polarity. Thats not to say some can't be reversed, I just haven't had that experience. I do agree that it should be first tested. If you have a variable power supply you could start at a low voltage like Erdin suggested and slowly bring up the voltage while monitoring how much current your using.
Best of luck!