Do they automatically turn off when the key is removed or do they set off the same beeping as the headlights when they're left on?

the oem fog light only comes on with the headlights on.. so if turned off headlights then fog lights would be off.. and if worried about leaving them on when car is off this would mean your headlights are on and would have beep if left headlights on and car off.

So, the fog light switch only works when the headlights are on?

Any ideas how to have some kind of running lights on the xB? I'd like for the headlights or fog lights to stay on at all times while driving without having the tail lights on. Is this possible?

there's always a way. but i haven't looked into the way the fogs are wired.

There's instructions here:

http://www.scionlife.com/forums/viewtopic.php?t=4569&highlight=foglight+wiring

Found by searching "foglight wiring".