Debian would be a nice candidate. Since that computer is so slow and doesn't have much memory to work I would suggest Fluxbox as a window manager since is lightweight and very minimalistic. As for p2p programs good options are ktorrent, torrentflux, utrorrent with wine or, if you have enought time to learn it, rtorrent with on of it's many web frontends. For an alternative to emule you could use amule. All these clients have web frontends so you won't have to sit always in front of that computer to see if downloads are finished. You can always use your browser from your main computer.
You don't need to run media players in that computer. You could set up Samba to share the downloads folder to the network or, if your other computers are linux boxes, you could set up NFS. On the other hand I don't know if it's a good idea if you want to totally sanitize your network from p2p programs. Maybe someone more experienced could enlighten us. In that case available media players are VLC, Mplayer, Xine etc.
You may want to set up SSH to manage that computer remotely with command line and VNC to manage it graphically.
Good luck