I have a little Debian box that I would like to use as a media center. I have used the Media Player Daemon in the passed, but I would like to know if Amapache would do a better job.
What I would like is for the Debian box to be hooked up to my speakers and I can control what it plays remotely, which MPD does great. I would also like to be able to stream the music on that computer to other computers in my house. Would Ampache satisfy me?
Also, are there any good web based bit torrent clients around so I can get my Debian box to download things in the background?

I have almost exactly the same setup, including the use of Debian as the server OS. I use Ampache for remote access (if I'm at a coffee shop and want to stream music from home to my local laptop), but log in via ssh and use moc when I'm at home and want to listen to some music on the server. Basically I use Ampache when streaming music from the server to a local machine, and use ssh and moc when I want to play music from the speakers that are hooked up to the server. You could try a similar approach, depending on how comfortable you are with the command line and console.

Not sure about the bittorrent question, but TorrentFlux looks promising for what you are interested in.