I think popping the garage roof up above the roof of the house would actually improve the look of this house. I would go up just a few feet and keep the same slope.
I can't help with a cost estimate. I'm sure some folks here could if we knew what part of the world this house is located.
I think that would look pretty cool, too.
I'm a residential construction contractor, which means two things: 1) I might be able to ballpark a decent number but 2) there's no way I'd do it on the basis of one photograph.
I'd need a lot more information. Raising the roof but maintaining the same slope would create a wall on the house side that would then need to drain effectively to the front or the back. How do you accomplish that without modifying the original roof? Same theory applies for the addition of almost any kind of roof (gable, hip, etc.).
Secondly, without more information it's impossible to know what exactly a roof-raising would entail. It isn't as simple as tacking a bunch of cripples on top of the current wall and slapping a roof onto them.
This isn't to say it can't be done; but it isn't gonna be cheap, and there's a decent chance you destroy the character of what is a handsome little home.