#1 and #2, NO!
#3. Once the water leaves the pressure valve it is no longer considered "potable water" and therefore should not be injected into the well.
4. Yes there is something else going on, but we cannot tell what without being there to observe the conditions when it occurs and test it.
Incidently, even if the check valve were partially frozen, it would thaw almost instantly once water started flowing through it.