Number of strings with unique substrings
You mean that each nonzero n-bit substring occurs once, and the all-zero n-bit substring occurs no times, right? Essentially, what you are looking for is a De Bruijn sequence.

The number of de Bruijn sequences should be in OEIS somewhere but I'm having trouble finding it. My guess is that it grows doubly exponentially, since it is the number of Hamiltonian cycles in an exponential sized graph (the de Bruijn graph) and graphs tend to have exponentially many such cycles when they have any.