Can my iTunes library be stored on an external HDD and work over multiple computers?

Yes this works beautifully.

I have all of my media (photos, music, videos) on a 500GB external hard drive.

My iMac is plugged into it (via firewire). Set the path as you describe (in iTunes AND iPhoto) and bingo.

Furthermore, my laptop connects to my iMac through my wireless network and accesses the same hard drive/media files.

And, my Apple TV downstairs in the living room is set up just like the laptop.

All access the same media files on the external HD.

Very nice!
What happens if the hard drive isn’t plugged in? Wow I’ve asked lots of questions about the same thing…

Yes, it’s pretty much the best way to do it. I’m home for thanksgiving and picking up more music on a hard drive to bring back to my place.

If the hard drive isn’t plugged in…it won’t get any electricity and won’t work.

Yep. Actually to save mounds of hard drive space on my MacBook Pro I don’t keep my library there at all. I manually manage my iPod content and everything is there on the iPod. Then I just back up my iPod to another external hard drive using iPodRip. I do a backup about once every 2–3 weeks.