Note that the morning star patty is vegetarian, but not vegan. Also it is microwaved and not prepared on the grill.
They're probably unwilling to dedicate the kitchen space for a vegetarian grill, so the microwave is the obvious choice. Microwaves won't cross-contaminate like a greasy grill will, and are already in the kitchen so the financial risk is minimized.
Also, many fastfood places do have dedicated grills for vegan/vegetarian burgers, like for example McDonalds in the UK.
Massively wrong. Most are easy going regarding using the same cutlery, pots, pan, etc (only ones I've met who aren't happen to be Indian mums or super strict vegans).
However... cooking up on the same griddle? Yeah, that's something most veggies and vegans I know who be very unhappy about.