I'm no engineer, but I can tell you this: a 2x anything isn't going to be enough. If you ever find a straight one that long, it won't be straight after a year outdoors. Once it warps out of it's plane, it loses its ability to bear a load, and can buckle. You can reduce this with blocking, but that gets ugly fast.
I would suggest some sort of engineered wood product, like a laminated beam. Maybe a 4x10? It will have to be watherproofed and occasinally re-coated. If you run 3 of those across the large span, you can then run 2x6s between them in the other direction (perpendicular to the big beams). They make brackets to hang the smaller wood between the beams instead of having to go on top and make it even taller. What about steel beams clad in wood?
You definitely want to run this by an engineer.. your side walls may not be strong enough for the load, especially in shear. Attaching it to the house will help, but without any sheathing on top you'll still have problems. You can help that by putting an X of cables across the whole top, but you'll have to make sure the ends are mounted well and that they stay tight.