Just type /quit
Both /query shark Hey shark, how's the beach? and /msg fox wassup dude work. In some clients, you can also click, double-click, or right-click on someone's name to begin chatting with them.
A more sophisticated variant of the same webclient is available here. There are also clients you can download and install. Both Pidgin and HexChat are popular and easy to use.
Yes. #broward is on anthrochat, an IRC network specifically for furries, with tons of channels on many topics. Another channel relevant to you might be #florida. Try /join #florida to join that channel, or try /list to get a list of all channels available. That's not all, though: there's not only one network for furries, but two! Furnet also exists, and also has tons of channels for furries to chat in. It's also possible to find a few furry channels on nearly every IRC network.
IRC is Internet Relay Chat.