Since you have a stick built roof, you have a few options to modify in place as others have suggested. However, unless you're planning to modify your garage door, there will b an issue with using a lift for vehicle storage.
A standard garage door at 7 or 8' should open above the lift which could work for pulling a car onto the lift, lowering the door, and then raising the lift to work on the car. But once the car is in the air, raising the door will hit the car on the lift. At 22' depth, you won't have enough room to push the lift forward to clear. You could go to a roll up door like they use on self-storage units or commercial warehouses, but you probably can't match the other door that way.
Hopefully I didn't miss where you already addressed this...
I also agree with others, I'm not sure the math works out. If you're 5/12 pitch, assuming you had no cross ties and vaulted the ceiling, just the rafters would be at 15 inches higher than your existing wall height 3 feet from the outside wall. If your existing ceiling height is less than 10 1/2 feet, it might not be enough.
Looking at the pictures, it looks like the garage is the only thing on that end of the house. In other words it doesn't appear that there is living space behind the garage. If that's the case, it should be possible, although still somewhat disruptive and expensive, to remove the roof over the garage, build the walls taller and add a new roof. Kind of the reverse of what you appear to have now, a step up in the roof over the garage...
