I am a gnu/Debian enthusiast. Here is what I can contribute from my end:
Debian has something called a netinst. Which will only install the core system from the CD ROM then pull the rest of the system from the Internet.
You don't have a CD ROM, but you can still follow the netinst method. I found this link:
http://www.debian.org/releases/stable/i386/
which will allow you to install good ole Sarge onto your system with using your floppy. Sarge is old, however you can find the testing branch called "Etch". I use Etch on my home server and workstation.
just be aware, this is a text base install...pretty easy though if you follow the prompts.
Post any questions and I would be happy to see what I can come up with.