3.5 mm to composite

The 5th Gen iPod output video through the dock connector not the headphone jack.

You need the "Apple Component AV Cable" or the "Apple Composite AV Cable"

Component is for HD tv and enhanced tvs http://store.apple.com/us/product/MB128LL/B
Composite is for standard tvs http://store.apple.com/us/product/MB129LL/B

Message was edited by: Sherman Campbell

Yes, the 5th gen, 30GB iPod video plays video thu the headphone jack and also thru the iPod connector on the bottom.
If you really want to get confused, see this -> iPod and iPhone: TV out support

The cable it just a standard "run-of-the-mill" cable which I found at my house

You will need to swap the yellow and red cables.
Yellow cable from iPod (right audio) goes to red input (right audio) on tv.
Red cable from iPod (video on iPod) goes to yellow input (video) inut on tv.

It's this way so you can use standard headphones with the iPod.
The standard "run-of-the-mill" cable you are using, uses different pins for the audio than headphones so it's color coded differently.

I purchased a non-Apple 3.5 mm to 3 RCA cable, which was specifically advertised as being for iPod-to-TV uses. I hoped to use it, as it appears to be designed for feeding audio and video from the iPod Classic's 3.5mm plug to my HD TV. Unfortunately, the iPod gives me the message "TV Out Enabled, Please Connect Video Accessory" and displays a picture of the bottom plug on the iPod, as if the bottom plug is the only route to feeding out the AV signals. I have set the iPod to TV Out, and tried both NTSC and PAL.

I assumed from the cable website, and the picture and description info that the cable I had purchased was designed just for my desired use. I think I've got the settings right, but I can't view the video. Did I order the wrong cable, or do I need to change my settings, or what? Every other function of my iPod works perfectly and I love it.

Did I order the wrong cable

Yes.
The iPod classic does not play video out of the headphone jack.
See this -> iPod and iPhone: TV out support