Will an attic fan in a garage also cool the adjacent house attic?

To ventilate a garage you want to exhaust air from inside the garage to the outside of the structure and not into the attic. The garage gets so hot that I think you should keep the ventilation separate from the rest of the house. Also there may be code requirements to separate the garage from the living space due to fumes.

If I were going to ventilate a garage, I would put in an exhaust fan in the ceiling or in an outside wall and have a fresh air inlet vent as far away as possible, on an opposite wall if possible.

Another possibility would be to have a power exhaust fan like a kitchen hood going out the roof and a vent coming in a wall or through a soffit to keep the inlet air as cool as possible. I wonder if the largest ceiling exhaust fan of the type designed for a bathroom could be vented out a soffit to avoid a roof penetration.

For example

It sounds like you're getting some heat radiated from the attic down into the garage. Insulate the garage ceiling to cut this radiant heat. Do you also have direct sun on a garage wall? Can you shade and/or insulate that garage wall?

If you put a ventilation fan in the garage, it should vent to the outside, not into the attic, to keep car exhaust fumes out of the house. You would also need to look at an intake vent from the outside into the garage so the fan doesn't pull the a/c from the house.

How would you activate the garage vent fan? A manual switch could work but you may forget to turn it off at night. Consider a thermostatic switch, like those used on attic (not normally on whole house) fans.