[Malibu79]

First off i have never seen any building in which the joists run opposite the rafters it throws off the whole purpose of the joists. It looks like it was an add on with either a retarted contractor or a previous owner who thought they knew what they were doing. Reguarding the project in mind it is usually possible depending on the hole desired compared to the overall garage size it is not a quick project or a simple one at that. It requires carefull planing about supporting the roof and keeping the walls from pulling out due to weight of the roof. Not to mention that u would have to get a permit due to resale and safety
if someone comes to look at ur house and sees major modifications they are not going to want to hear that "yea a couple of my friends and i did it and we are almost positive that it is structurly sound" as apposed to having a written document saying that the work was checked and perfectly sound

so i am not saying that it is not impossibe but it not something to be taken lightly alot has to be done to pull it off