Older:

Apples store best at 36-38 degrees in a cooler to keep them crisp. I sometimes store a bushel of windfalls on my patio (covered and out of the sun) and I can hold them for a week until I get my applesauce made.

Since you plan on using them for cooking and not fresh eating you should be fine. The apples will get softer after 10-14 days in non-refrigeration but not an issue if you are cooking them up.

For fresh apples I store in refrigerators but many of my apples get used for sauce/pie and they get stored in the milkhouse. I normally get many bushels and I do not have refrigerator space for them all. I try to use them up by thanksgiving for cooking purposes only once they get soft. Usually the last of them get made into sauce and canned up.

For your garage I would keep a window open at night to let in the cold air and then close it off during the day to keep the warmer air out. Assuming the days are warmer than the nights.