Are variable-length crypto hash functions still susceptible to collisions?
Is there any work done to show or prove collision resistance gained by increasing digest length?


Actually, as CodesInChaos has mentioned, the variable length versions of Keccak ("SHAKE128" and "SHAKE256") are known not to have any collision resistance beyond their security level, independent of how long we make the output.

So, what's the point?  So, as you ask:


  What are the other benefits of allowing variable and arbitrary length output?


That's because collision resistance isn't the only property we want from a random-looking function.  One such alternative use is a Key Derivation Function; this is a function that takes as input some secret data (e.g. DH shared secrets) and some public data (e.g. nonces) and converts that into session keys; these are a series of encryption and MAC keys (and as we might have several such keys, and they're of variable length, this set of session keys is long and of variable length).  Now, a number of KDFs are in use by real protocols (such as TLS), however these constructions are ad hoc, and actually rather ugly looking.  It would be nice to use a single function that's designed to do that; a "variable length hash" (or XOF, that's what NIST dubs it) may be cleaner.