There are fire code requirements for attached garages, but it should be doable. You will probably have to sheath the garage ceiling in 5/8" sheetrock at the very least. Most definitely air seal well between the garage and upper level. A ventilator system would be a good idea, perhaps one on a timer so it runs for a few minutes after you have closed the garage door and driven away in one of your hot rods

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The building official may spec a sprinkler system...
How exactly do you get to the upstairs game room? If it is THRU the garage portion, you may need to plan on an additional external door and stair to provide another means of egress from the upstairs area, or just make the upstairs access all external.
A few minutes discussion over some basic sketched drawings of your idea with your local building department might really help to get you proceeding in the right direction toward a set of plans they will pass without need for major revisions...