All stars have some angular momentum. Conservation of angular momentum when the core of a star collapses means that it is very likely that most black holes are rapidly rotating, just as newly-born neutron stars are observed to be extremely fast-rotating.
Even if one were to have a star with a non-rotating core as it collapsed, the subsequent accretion of material with angular momentum would give the black hole that angular momentum.
Non-spinning black holes are a theoretical ideal that is unlikely to be found in nature.
The details of the accretion of material onto a black hole are complex and still the subject of theoretical and observational research. But yes, the details of accretion in spacetime governed by the Kerr metric appropriate for rotating black holes will differ from those of a non-rotating Schwarzschild black hole. Kerr black holes have an "ergosphere", outside their event horizons, where material is force to co-rotate with the black hole.