I never warm the car before any drive. I just give her a few seconds to get the oil around, then off we go. I'd advise against idling for extended periods.
I'd guess you just have too much air in the system. Might be a bubble, leaking radiator, expansion tank seal, pressure cap, etc. You can test the engine fan with say ten pages of a newspaper. Turn the motor on...roll the newspaper up & stick one end in the fan. When the fan clutch is cold, it'll have trouble with the paper. When the clutch is warm (locking the fan up), it'll shred it. I wish I had a more scientific test, but that'll do for a quick result.
If you don't see any obvious leaks, I'd drain the coolant (both radiator and block drains) & do a pressure test. That will tell you a lot about the condition of the system. 15psi/1bar is all you need.
Frozen coolant isn't a good thing either, even a slush. I wouldn't think it did damage, but water does expand as it freezes, hopefully it only stretched the hoses a bit.