How does it look for my ailing EeePC ?

Not good, unfortunately.
It seems now that the principle problem involves the black-screening when the DCIn cable is plugged in or unplugged. This problem is still present after three days and is definitely not right.
My own feeling is that this has somehow messed up the Xandros OS (or could it have been Xandros' Grub?) so that nothing othr than Xandros is available. No Grub menu is available and no menus result from holding the ESC, F2 or F9 keys during the boot.
Essentially no replies were received to my forum post on this issue which generally means that the problem is not resolvable.
I sent a private message to what seems to be a key EeePC developer from the forums so it'll be interesting to see if any reply is received.
However, the most likely outcome is that this is a hardward problem which requires the machine to be replaced.

Yes, we're dual-booting on the EeePC

Once I had managed to get the new partitions on the 16 GB drive, installation of Ubuntu eee from a USB key was relatively easy.
The same precautions are required every time you want to boot from anything other than the internal HDs as the Boot Device Priority setting is not remembered from boot-to-boot.
Interestingly, the /dev/sdb3 partition that wasn't seen initially by Xandros presented no such problems to the Ubuntu installer.
The install went smoothly (without any of the "no-migration-assistant" problems experienced with Intrepid Ibex). Note that I chose to install no bootloader.
he problems started when I tried to configure the /boot/grub/menu.lst of Xandros to take care of the Ubuntu boot. I first tried this from within Xandros. Everything seemed to go well but when I rebooted (after having commented out the "hidden menu" line in /boot/grub/menu.lst), no grub menu appeared no matter how many times I tried it.
I eventually found what the problem was when I stumbled on this guide (see the section titled "Changing the original GRUB menu"). It seems the changes I had made went to /dev/sda2 while the only changes that will have an effect must be made to /dev/sda1. This device can only be accessed from another partition.
So, I had to go into a LiveCD version of Ubuntu eee (from the bootable USB key) and ad this line to the /etc/fstab

/dev/sda1 /media/sda1 ext2 defaults 0 0

Then I created the sda1 folder in /media and mounted it. Then I could cd into that folder and edit /boot/grub/menu.lst.
Having done this, it was also necessary within the same folder to store the vmlinuz and initrd.img files of Ubuntu eee to /boot/ubuntu. It seems that /dev/sda1 views the 16 GB HD as an external drive.
With this done, I now got a grub menu on starting up and could boot into Ubuntu eee by choosing it from the menu.
Ubuntu eee worked fine and, after adding 36 new updates, I was able to install Opera 9.60 (note that this time, libqt3-mt was downloaded and installed by Opera itself rather than me having to do it as was the case with the LiveCD version).
I installed Yakuake from the repos and Livestation from here. Livestation works pretty well bu Yakuake doesn't seem to like being at 50% width as pressing F12 drops down the 50% width consle but a "shadow" console also drops down in the remaining 50%.
I aso tried 3D effects but, although, I got the wobbly windows and emerald themes, I couldn't get the cube to rotate. I have feeling that /etc/X11/xorg.conf might need to be modified.
All in all, however, this was a major stpe forward and looks very promising.

Adding new partitions to EeePC 901

To get another Linux OS installed on my EeePC 901, I needed to add two new ext2 partitions to the 16 GB drive. So, I used Unetbootin to prepare the bootable USB key with Parted Magic installed on it. Note that Parted Magic didn't work for me on the EeePC due to the version of glibc being older than 2.4 so I had to do this on another computer.
To get the USB key to boot the steps are as follows:
1. Plug the key into the operating computer
2. Reboot and hold down the F2 key until you see the TTy1 screen show up
3. When the BIOS screen is available, tab over to Boot
4. Arrow down to Hard Disks -- now you should see two HDs followed by a USB Flash drive
5. Use the controls (the '+' sign) to move the USB Flash to first position
6. F10 to save and exit

Now you should be on your way to booting from the USB key
However, in my case at least, I kept getting kernel panics. The error given was

EXT2-fs error (device ram0): ext2-get-inode: unable to read inode block

In googling around, it seemed this problem may be due to ramdisk_size being too small.
So, on the next boot into Parted Magic, I used the Tab key to edit the default option. As it happens, the last entry in the command line was

ramdisk_size = 25000

So, I just changed this to 250000 and booted.
Now it started fine and I was able to use Gparted to generate the needed partitions on the 16 GB drive without any problems.
When booted back into Xandros, I found I was only able to mount one of the two new partitions. I posted to the forum about this and it turns out that even though the unmountable partition /dev/sdb3 had been generated in Gparted, it still needs to be created when back in Xandros.
I used the mknod command given in the post and this worked fine.