2 things I thought about. One is the pitch of the roof. And the other is your 4x4's you said 3 on a side (corner, middle, corner) and the run is 12x15 so the distance between the 4x4's will be about 6' or 7.5' and thus need some good support (probably greater then 2x4's). Your roof is gonna take almost 6sheets of osb or plywood, tar paper, then the asphalt shingles (that's a decent amount of weight)...living where you do, ya might want to add a gutter and a rain barrel. Not sure how much you want to spend but I think id go to the pro desk at he depot and tell them what your making. They can tell you what size of 2x you'd need for that span. I am by no means a construction guy other then my own crap. Anyways that just my 2cents.