The badges can be used to allow your Pokémon to use certain moves outside of battle. The Zephyr one in particular will allow you to use Fly - assuming you have a Pokémon that can learn it, as well as the appropriate item to teach them.

As for the comas, they are reasonably common. They seem to occur for about a week, and the one in the coma usually awakens after having spent some time at home.