1) if it will never contain a hard disk there are a couple of options...
first live cd, e.g. knoppix. boot to that and use a floppy drive to store any configs for your personalisation. you could also share a space on your main machine for a /home for this diskless machine.
alternatively an nfs mounted system, where you have a fullly fledged linux distro running via an nfs server. i've written up my version of this here:
http://mini-itx.com/projects/animalsnes/ although i'd not expect your machine to be PXE enabled, in whcih case you'd need a syslinux boot disk instead.
2) not a question
3) personally i found gentoo extremely easy to install onto nfs as there's no real installation program. once you've mounted a filesystem somehwere, you just splat the base layout of gentoo onto it, and it's done. other more convetional distro's should be usable too, but i'd expect it to be harder.
4) also not a question
5) ok.. that's a 40% question ratio... not good!