Can stream ciphers (usually) be "run backwards"? Is it possible to reconstruct the previous output bits of a stream cipher, e.g. RC4, when only the current state is known, or is that computationally hard, or even impossible (due to ambiguous preceding states)?

For Salsa, it's obviously possible, since that even allows for random access.

Is there a formal name for the presence or absence of that property ("backward secrecy")?
With RC4, the answer is "yes, you can efficiently run the cipher backwards, reconstructing previous states".

For stream ciphers, whether you can reconstruct previous states in not typically considered, however for cryptographically secure random number generators (which are a similar primitive), it does come up; the term I've heard is "Backtracking Resistance".

Now, there is one thing to be careful with "Backtracking Resistance"; if the reason you have backtracking resistance is because the function that transitions between the previous state and the next state acts like a random function, then it becomes easier to fall into a short loop than if the transition function is invertable; this is because the you run into a loop if the next state happens to be any of the previous states you have visited.  In contrast, if you have an invertible next-state function, the only state that you might revisit is the initial state.  Because of this increased tendency to fall into short loops, it's typically considered wise if you (at least) double the size of the internal state beyond what the security strength might indicate you would need.