G4 mac mini won't start-up in Leopard anymore...

the symptoms are pointing to the update not being completely installed. as in something happened during the update. i suggest taking the leopard disc that you have and doing an archive and install. this involves the computer installing a completely new copy of leopard right over the existing one and preservers all of the user data such as documents and music and stuff. and is very easy to do.
just stick in the leopard disc and when you get to the hard drive selection page just select the external and click options and select archive and install and click ok. then just follow the rest of the instructions.

That model of mac mini should run Leopard perfectly well, even with 512Mb RAM, so the question as to which OS to use ought really to be a matter of personal choice in terms of which has the features and software compatibility that is most useful. There is also no obvious reason why there would have been a problem with the Leopard installation on the external drive, though it is certainly possible the update didn't correctly install. Since I run 10.5.4 on two Mac minis of broadly the same specification, it's not an issue with the OS.

The first thing to try, I think, would be to download the 10.5.4 combo updater from http://www.apple.com/downloads/macosx/apple/macosx_updates/macosx1054comboupdate .html This is a large download, but these 'combo' updates commonly resolve many of the sort of issues you describe.

If that doesn't work, then I think it may be wisest to reinstall Leopard from the install disk. Preferably this should be a full erase and install in order to ensure the drive formatting and catalogs are correct, but if there is user data on this drive that can't be (at least temporarily) moved to the internal or another drive, then macboy's suggestion of an archive and install is next best. Beware that by default, the Leopard installer will tell you that you can't install leopard on the external drive - this is because the copy already on there is newer than the one the installer can create, and is not telling you that there is any other issue!

A failed FW port is often a symptom of a power management failure in the system, and can commonly be resolved with a PMU reset:
To reset the PMU
-Unplug all cables from the computer, including the power cord.
-Wait 10 seconds.
-Plug in the power cord while simultaneously pressing and holding the power button on the back of the computer.
-Let go of the power button.
-Press the power button once more to start up your Mac mini.