Short answer: no, there is no fixed frame rate or frame-based processing in mammalian vision.
Photons arriving at the photoreceptors at the back of the human retina interact with photo-sensitive pigments called opsins, and modulate their release of the neurotransmitter glutamate . The level of glutamate released from a photoreceptor then changes the membrane potential of the other neurons in the retina connected to the photoreceptor (bipolar cells, for instance). Signals are processed within the retina, then passed through retinal ganglion cells (RGCs), along the optic nerve, through sub-cortical structures and in to visual cortex.
This process occurs continuously, but all of the elements along the way have intrinsic time constants and other temporal constraints (number of photons required for opsin activation; recovery rate of the opsins in the photoreceptors; membrane time constants and refractory periods of neurons; synaptic delays; etc.). Persistence of vision (the effect underlying the perception of a movie as a continuous stream rather than a series of distinct frames) probably relies on the conjunction of many of these time constants.
Concretely, a flashed visual stimulus produces a response in (primate) visual cortex after around 30–50 ms (Maunsell and Gibson 1992). You can recognise and subsequently recall a lot of detail from a sequence of images flashed at around 100–150 ms per image . These are still not hard and fast definitions of the "speed" of vision.
Since we only learnt to produce movies as sequences of distinct frames, it's tempting to guesstimate the speed of human vision in those terms. Biology, as always, is more complicated.
As far as the "spinning rims" issue goes, there are many aspects of cortical and subcortical activity that tend towards oscillatory activity. In a system with an intrinsic frequency (for example, the neurons in the thalamocortical loop between LGN and primary visual cortex), providing an oscillatory stimulus can synchronise the system to the stimulus.
Many of the intrinsic time-constants I described above could result in a tendency to oscillate at some intrinsic frequency, and then be subject to phase locking due to a periodic stimulus close to that frequency.