Cold will not damage primers, propellant or loaded ammo but, as noted, may affect the performance of the latter (if fired v cold).
Heat on the other hand, can be very damaging. It speeds up what amounts to a normal (albeit very slow) chemical decomposition process. If your stuff is still out in the garage next summer and it gets very hot, it would not be good for it.
Moisture is not that serious. Take a round, drop it into a glass of water, wait a week and it will normally fire. If it can handle that, what do you think a bit of airborne moisture will do? If the garage is not soaking wet, there should be no need to take extraordinary steps like wrapping it in plastic.
Ideal storage, of course, is dry and cool.