Is there an FPRAS for the number of min st cuts in general graphs?
The Provan and Ball paper actually shows a 1-1 reduction from bipartite independent sets (BIS) to min st cuts. So an FPRAS for min cuts would imply an FPRAS for #BIS. However, #BIS is not known (and thought unlikely) to have an FPRAS [1][2].

[1] Dyer, Martin, et al. "On the relative complexity of approximate counting problems." International Workshop on Approximation Algorithms for Combinatorial Optimization. Springer, Berlin, Heidelberg, 2000.

[2] Dyer, Martin, Leslie Ann Goldberg, and Mark Jerrum. "An approximation trichotomy for Boolean# CSP." Journal of Computer and System Sciences 76.3-4 (2010): 267-277.