Does this place have a vegetarian menu or known vegetarian options? Do they, or will they, cook eggs/omlettes in a pan or grill where meat is not being (or has not been) cooked?

You will have to ask them. Staff was friendly

Doubtful. They got upset when my wife asked to sample the chili.

Doubt it. The have vegetarian options but it all gets cooked on the same grill/pan as meat.

Unsure