To reply to a chat by GMail, find the chat in question under 'Chats' in
your GMail folders, scroll all the way down and just start typing.

I don't know much about using GMail with local storage and search, but
I can tell you that it is possible to download your messages to a
regular desktop client using POP; I also seem to remember that Google
Desktop has an option to let you search your GMail. A locally installed
GMail server is not going to happen, though: I very much doubt that
Google is going to make it available for free, and even if they did, I
have to assume that the hardware requirements would be higher than most
home users would be comfortable with.