The Asus EeePC is a great platform for tinkering, but several users reported to revert back to the original Xandros version for the machine, mainly due to the space other OS-es take up in comparison with the customized Xandros Asus deliver with their product.
1) Debian takes as much space as you'd prepared to give it. But for a complete distro matching the default Xandros, assume no less then 1.5 GB.
2) the net-install iso would be a good start
3) UK magazine Linux Format did a series of articles about the EeePC, dealing with this: http://www.linuxformat.co.uk
and have a look around on their forums too.