Yes, they have vegetarian options! Check out the veggie soondubu.

If you are just gluten sensitive (but not celiac), there are some dishes that are possibly GF/low-in-gluten.. Like the rice cakes, japchae, rice, meat and veggies. I didn't check if their soy sauce is GF or not. We did not have any bad reactions when we went.

No, they don't have patbingsu menu at the moment in 2018.