Will it damage ammo to be stored in very high or very low temps, or widely varying temps, such as in a car? Will it matter if it's pistol or rifle caliber? Thanks!

I have ammo that has been in my trunk of my car everyday for years.  Weather goes from -30 to 100's in the summer.  It all goes bang.  heat will degrade ammo more then cold but I wouldn't sweat it.

I would just switch out the ammo yearly for duty/carry ammo that way you will know that it will go bang when you need it to.

Speer LE
Ammunition Storage Recommendations:
Ammunition exposed to temperatures above 180° F can cause the propellant powder to deteriorate.
Ammunition inventories should be stored in a stable climate, not subject to temperature fluctuations that would facilitate water condensation. Optimally, an indoor facility with climate controls is best

SOURCE:  le.atk.com/pdf/AmmunitionStorageRec.pdf