UPDATE:
Every major part of this system (other than the air intake hole on the opposite side of the garage. I’m just using the big door cracked open slightly for now until I have time for the hole project) is now in place, and functioning as expected.
When the outside temp is x-degrees lower than the temp in the garage, it turns on the fan.
Since there is so much heat built-up inside the garage, in the walls, the ceiling, the concrete floor, and all of the other objects in the room, it will probably take a few days to really get to the point where I need to be, but even with just the first night of having it in place last night, I can already see the difference it has made…
Normally, in the summer, it is so much hotter inside the garage than outside, that it never inverts by itself (i.e. hotter outside); it builds up so much heat throughout the day with the sun beating on it, and even though that heat then radiates out into the house all night long, it never loses enough of it from the structure of the garage to get to the point of it being hotter outside.
Today, we finally saw the inversion take place… After leaving it run all night last night (taking cooler air through there all night long), I think it really worked on all of that trapped heat in the structure of the garage.
At one point today, the fan turned off. When I checked the devices/temps and the SmartApp, I saw that the outside temp was higher than the temp in the garage (in the summer, this has never happened at any time, day or night, in all the years we’ve lived here; so, substantial confirmation of the general working premise here), and that was why the fan was off. The fan got the heat out, and the SmartApp noticed and turned the fan off. 
The whole reason for doing this in the first place is to hopefully save money on electric bill by reducing the amount of heat that the AC unit has to remove from the house simply because it’s radiating in there from the garage and the attic. I know the heat builds up gradually, day after day, and there is a cumulative effect (terminology=?). Hopefully, it works at least similar in the other direction, and after a few days of extended use, the saved heat will be significantly gone. That way, this new fan doesn’t have to keep running all night, every night. A while of this at the beginning makes sense, but I don’t want to just trade one waste of electricity for another.
The fact that the fan turned off in the middle of the day is a really, really good sign that this is actually working. I just went out into the garage, and I was amazed! Granted, it’s still way hotter than it is in the house, but being cooler in there than the outdoor temp is incredibly encouraging.