You can still install Slackware from floppies, but I don't think you'd want to! As for the ethernet install, yeah - just about all distros support this one. Slackware via NFS, Redhat does NFS and FTP installs. Dunno about the others. Have a look on
http://www.linuxdocs.org for relevant HOWTOs.
Personally I would 'lend' the machine the CD-ROM as you put it. Probably be the fastest route at the end of the day. With the HDD size I would look into one of the distros that is very compact, say Peanut Linux, but there are others...
HTH
Jamie...
PS As its an old machine it probably won't support booting from the CD so remember to make your boot disk(s).