The first remedy to use is Disk Utility from your OS X Panther installation CD, not Disk Warrior.
Your PRAM battery is almost certainly depleted, resulting in corupted PRAM. The battery is easily replaced, but you really ought to be able to boot after the computer has had line power for a few moments before attempting to start it.
Try another PRAM reset. Do this two or three times, followed by booting from an OS X installation CD and running Disk Utility.
It may be necessary to reinstall OS X if its copy has become corrupted. Doing so will not erase your user accounts or data.
There are two possible reasons the Flash Drives don't work:
1) The Flash Drive is probably not recognized because it is formatted in something like MS DOS Fat32, and your old iMac doesn't recognize that format.
2) Most modern Flash Drives are USB 2.0, and I don't know if they would work on the old iMac that only has USB 1.1. I have an old 100 MB USB 1.1 Flash Drive I keep for use with my (hardly ever used anymore) Clam Shell iBook, because of this (I think).
If it's a format issue, you'd need to find someone to format the Flash Drive to Mac OS Extended. But if you're wanting to use the Flash Drive to transfer files to a Windows based machine, I don't think they recognize Mac OS Extended. Daxxed if you do, daxxed if you don't. (Insert the appropriate letters for the xx's, I wasn't certain the protocols on language here in the forum.)
Try removing the preferences files for Safari, iTunes, Open Office, etc., and restarting those programs. they might just have bad preference files.
Were you originally running OS 10.3.9, and are you still runing OS 10.3.9 after the OS reinstall? Your OS 10.3 install disk might have been a earlier version of the OS 10.3 you were originally running. If you're not running 10.3.9, you might want to try a software upgrade to 10.3.9.
I trashed the preferences and restarted. ITunes still will not open at all. Safari, though, now opens to a blank window and goes no further.
I don't remember what version of 10.3 I had. What I have now is just 10.3. Per "About this Mac," there is no further digit on the number. Software Update says that my OS is up to date.
Return the Preference files to the original places.
Here's a link to the OS 10.3.9 combo update (updates all versions of 10.3 to 10.3.9):