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