Hot and cold do measure motion, if we want to be very technical (since temperature is just a measure of the average momentum of molecules).
Pain should probably be considered a separate sense indeed, though it can also be argued it is something different from the senses themselves, since it is fully internal, it doesn't reveal some fact about the outside world.
Trigger warning: description of medical procedure.
Yes pain seems more like the interpretation of sense data, like if something sounds nice or tastes good?
When I had an operation under local anesthetic I still felt the knife cutting into me, but was able to interpret it as not painful? This is how opiates (for pain relief) have been described to me -- still allowing you to feel the sensations but allowing you to not 'feel' the pain.
Perhaps seeing with human biology knowledge can tell me why this is wrong!?
Pain should probably be considered a separate sense indeed, though it can also be argued it is something different from the senses themselves, since it is fully internal, it doesn't reveal some fact about the outside world.