+1 on Birch plywood. I have pieces (circular and straight) for elevated track on seasonal layouts that get stored in the garage. No warping over several years of crappy summer storage in mid-Atlantic heat and humidity. Cannot say the same for softwood ply. I would not touch MDF as it needs support on all edges to retain its shape over time.
I concur with leaving the outside corners on for stiffening.
To support a long bridge, use angle or square tubing under the bridges. I find that aluminum angle with 1/8" walls is stiff for supporting bridges. You might consider aluminum U-channel of a width that matches the underside of the bridges - it will be very stiff even over long spans. Check out onlinemetals.com for available shapes.
