Live CDs might be a little dicey on a Pentium I with only 32 MB ram (dicey or impossible). But you can certainly try the Live CD installers for, say, Ubuntu on your desktop to try them out there.
Try Xubuntu. It is Ubuntu with XFCE window manager, rather than Gnome, and a few other lighter weight packages as default. You get all the benefits of a Debian-based distro, in a lightweight flavor. They should have a Live CD installer, so try it on your desktop to see if you like it.
You could also just install a minimal Ubuntu and choose other lightweight DE/WMs. Noone says you have to use Gnome with Ubuntu.
For application installations, the best bet is a Debian-based distribution. Apt is by far the best package management solution, in no small part thanks to the massive repositories Debian (and others, like Ubuntu) maintain. Redhat's RPM is a failure due to lack of a good central repository, IMO...
Any application should run on any distro of linux; if they don't provide an Apt package (or RPM for Redhat, or a Suse package), you will likely have to compile from source. Its usually not too difficult, but it can be intimidating at first.
Wesnoth at least seems to have binaries in the Ubuntu and Debian repositories, so it will be a snap.
Open Office is in everything too, though I dunno how it would run on the laptop.