Why are the metal contact points shaped like Ts? Is the relay supposed to sit slightly above the board on the wide part of the Ts (only the narrow parts go into the through holes) for airflow purposes?
The upper part of the "T" shape is there to prevent pushing the pin right through to the other side of the plastic bottom. When you go to stick the relay into a board, there is some pressure against the pin because the fit isnt perfect as the pin rubs on the side of the hole. If there was no upper part to the pin the inside guts would have to bear the force instead which could damage the inside parts, or even just push the pin up inside the casing where it would no longer be accessible for soldering.
Obviously the pin part is made for soldering on the foil side of the board of a single sided board for example.
That depends. The relay is rated for 40 amps. If you are using it anywhere close to it's rated capacity, you will need some wide, heavy traces on your PCB. The stepped down pins allow you to solder it to a smaller hole, but you should have a large bare copper area for the shoulders to rest on, with some additional solder to help carry the current. This, of course, implies a two+ layer PCB with plated through holes.
You can get a better solder job with the relay raised off the board, more flow on the top side. The solder on the top should flow the hole width of the pins on the relay.
Use a small hole with wide traces.