Well, the only viable third faction I can think of would be(warning Imperial Agent spoilers soon) is the
. The Hutts are powerful and rich enough to become a third faction, but I think they'd need to ally with some other group(s) to be viable enough to stand toe to toe with the Empire and Republic.
New classes, I'm not quite sure because BW did a good job of taking iconic classes or characters and representing them as a class or AC. I'm not a huge fan of the typical suggestions that crop up in threads like these, from former SWG fans that often suggest playable classes from that game for TOR because they just don't have the right feel for TOR . I do like the Tera Kasi idea and a few others, but I really haven't thought enough about it for my own suggestions.
EDIT: I think someone mentioned the Imperial Guard from this era,they would make for an interesting class. They seem to be a mix of melee and ranged, so you could give them a tank,dps and possibly healing spec.