Is the open grate in the ceiling between my garage and attic necessary?
Your attic definitely does need ventilation.  However, it seems really odd to me that it is vented down into your garage.  Is there any kind of gas powered appliances nearby?  If not, I would:


Install some roof vents and maybe an attic fan.
Install soffit vents or some way for air to get in.  (you want to create a FLOW all the way through your attic)
Drywall over the vent in your garage.