[COLOR="DarkRed"]do I need to replace the heating elements?[/COLOR]
Yes you will have to replace the heater assembly as it is grounded.
Even if you could un-ground the coil once they overheat/warp they are weaker so odds are it will just happen again at a lower temperature.
[COLOR="DarkRed"]you'll see the left lead on the cord has started to corrode and the insulation partially pulled away.[/COLOR]
It looks like it has over heated and has started to melt.
The power cord should also be replaced.
[COLOR="DarkRed"]Ok. How would I check the motor with the multimeter?[/COLOR]
It is a bit of a pain. In order to check you will have to disconnect the 6M and 4M wires. Make a note where they go as you do not want to reconnect them wrong. Then measure the windings to 4M, see the wiring diagram. Them measure from the connections to the motor case all should be open (infinite ohms), this is to see if a winding has shorted to ground.
This just tells you that it is OK electrically, it may still have a mechanical problem.
Rotate the armature to see if the bearings are OK.
[COLOR="DarkRed"]OK, I checked the upper thermal fuse. Which others should I check? Do I just check them for continuity to see if they're good?[/COLOR]
Check them all. They should all measure 0 ohms.
[COLOR="DarkRed"]I can do the cleaning of the chassis, blower wheel, etc. easy enough. Which specific parts should I order to give it an overhaul (belt, rollers, etc.)?[/COLOR]
Check the roller's bearings and their shafts for wear and slop.
Check the belt for cracks.
[COLOR="DarkRed"]
Another data point: I did NOT get continuity across the thermal fuse (thin, white piece of plastic attached to the back of the blower wheel cover). Could that be the source of the problem? Or, is it just another symptom?
This is the fuse that will stop the motor from running.[/COLOR]
Cannot say 100% why it blew. Could be that the grounded heating element got so hot that it blew the fuse, could be your clogged vent.
If you have to replace the motor do not be surprised if you also have to replace th blower wheel, Often they seize onto the motor shaft and the only way to get them off is to break them.
http://www.applianceaid.com/wpmotor-change.html