A commerical "cold pack" consists of a bag within a bag.  The inner
bag contains salt, typically ammonium nitrate. The outer bag contains
water. Punching the bag breaks the inner seal, which allows the
ammonium nitrate to mix with the water. As the ammonium nitrate
dissolves, heat is absorbed, and the temp. of anything is in contact
with the pack - including sprained ankle. Is this an exothermic or
endothermic process?  Are the changes physical or chemical?

This is an endothermic reaction.
It is a physical change. You can separate the water from the salt by
simple distillation to obtain the dry salt and water. The two
materials have not been changed.