Wii doesn't show up as a console to play on for runs, so does that mean Wii isn't allowed for runs, or is wii allowed and there just is no other console option?

Wii is allowed using a Gamecube disc. They just didn't bother to add the different platform when submitting.

Some games track by platform played on, some track by platform of the game. TWW tracks by game platform; i.e. all runs are GCN version, regardless of what it's played on. Other games maybe not so much, but at least for TP and TWW this is how it is.