No need to book to visit temples, you can just show up there. Booking probably needed if you want to spend the night, in case you are interested, you can check out http://eng.templestay.com/
About vegan food.. don't really know but I remember going to a very good kimchi place where the tofu is really nice and you can have soups or other dishes with it, just check before cause some dishes have meat, it is called Blue Star, close to Anguk metro station.
Also, be aware that if you ask for no meat, they may just pick the meat out, leaving the meaty sauce/broth behind.
Check out the Gwangjang Market. It is the old, central market and a labyrinth of covered alleys with very traditional vendors. In one part you will see the women making fresh bindatteok (mung bean pancakes). Sit at a place where you see them grinding the mung beans. Cost is 4,000 won for one really big pancake - enough to feed two. Many other jeon vendors where you can pick out your ingredients, like mushroom, peppers, or tofu.