Do they cook the omelets using milk with the eggs?

No

I can't be certain but I'd be willing to bet yes since that's standard, and they s pretty traditional eatery. But who knows, they could have egg substitute if asked. Just give them a call. Good luck, my whole family loved that place... It's out go-to breakfast spot.

Not sure but you can ask. If you're staying away from milk I'm sure they would be more than willing to accommodate you. My favorite is a 2 egg veggie omelet with swiss and hot sauce. Enjoy!