Does mutt include your sent mail in that view? This is one of the gmail features I love. I don't have to dig around to see the flow of a conversation with gmail.

Also, if I receive a message, reply to it and archive that thread it goes away. But if that person reply's again it brings the whole conversation back to the inbox so I can see the previous messages as well.

As someone who uses the inbox as a partial todo list that is amazing. It makes it difficult to switch from gmail, although I've been using mu4e[1] inside emacs.

[1] http://www.djcbsoftware.nl/code/mu/mu4e.html

To include your outgoing messages in threads, just set up a hook that adds an Fcc: +inbox header. I used to copy all outgoing messages to a separate +sent folder, but that didn't turn out to be very useful.