Does Magma burn through squares and walls and floors etc? If I'm going to build a lava resevoir, do I need to make sure the entire magma channel and the reservoir is all completely magma safe to keep it from spreading?
Basically, no. All natural and constructed walls, floors, ramps, or stairs should be okay. When adjacent to or submerged in magma these structural elements do not melt or catch fire. Although, I believe there is an exception for natural ice, which plays by special rules to enable the seasonal freeze/thaw cycle (which remains true even in glacial biomes without seasonal melting).
Any pumps, doors, floodgates, bridges, hatches, grates, bars, etc. that may be along the path should be made of magma-safe material if you want them to last. This includes every mechanism used to connect to a lever or other trigger as well.
drawbridges when they're up?
This gets tricky and is not dependent on the up/down status but rather on exactly where the magma flows. All materials used in the construction of the bridge, as well as any connection mechanism(s), are treated as if they are located in the original centermost tile for temperature/destruction checks. (The bias is toward the top/left in case of even-numbered spans.) If for some reason magma submerges that central tile being raised will not save the bridge. This is also true of retracting bridges: being "retracted" and not visible is very much not the same as being magma-proof.
Note that in the case where the x or y dimension of the bridge is one tile long and it's set to raise correctly, the central tile will be underneath the raised "wall" and thus protected—as long as the bridge remains "up" of course!