You could accomplish most/all of this using an Arduino with an ethernet shield, or the special Arduino Ethernet board. The newer ethernet shields even come with a micro-SD card slot. The BitTorrent protocol should be simple enough to run on a small device, though I'd probably invest in one of the more powerful boards (e.g. Mega2560) to be safe. You can read from and write to FAT16 and FAT32 filesystems on standard SD and SDHC cards using the SD library. They've even been nice enough to provide libraries for Hitachi HD44780 compatible screens and another for KS0108 compatible screens to take care of the display part. There's also a great SHA1 library for Arduino, which will be necessary for computing info-hashes and verifying data parts.
Your part of the project would involve implementing the bencode routines for tracker communication and communications protocols for client-to-client transfers, then tying it all together.