I don’t have any good examples that I know of, but I am very curious. To me though, a rivalry is a rivalry when two or more of people are closely competitive. But is it really a rivalry anymore when someone has more victories over the other? I don’t think so.
Yes*.
Technically speaking, there’s the Green Party and PSL on federal ballots for many states, but the electoral system is positioned so that there’s no foreseeable way that a party other than the two main ones could realistically win. So for all intents and purposes, there’s no federal left-wing option.