Welcome. And such bad luck
I my experience, installing drivers from HP has always resulted in (big) trouble, be it on OS X or on Windows XP. Most probably Tiger has your printer driver pre-installed so you don't need to run the HP installer.
But reading on, it seems you might have a lot more problems than can be accounted for by a printer driver.
I suggest you do a clean install, run software update once you have it running, and start all over. If you run into problems again, I'd return the machine.