I just connected my iPod to my computer. Can I delete songs from it by just undoing the check-mark next to a song title?
Yes, though you have to re-sync the iPod for the deletion to occur.
I think I’m one of the few that never syncs my iPod and always manually manages what goes on and comes off. What I do is highlight the song on the iPod in iTunes and just hit the Delete key. It asks me for confirmation then deletes the song from the iPod.
Just did it by hand, and it worked!
That’s what I do too, Shoeless. I got an Ipod shuffle for Christmas last year and while I love my IPod, I have a love/hate relationship with ITunes at the moment.
This. If you just uncheck it, it won’t be played, but it’s still there.
iTunes asked if I was sure I wanted to delete everytime I deleted a song.
You can use winamp to add and remove songs from an iPod.
That depends. I have my iTunes set up to ‘only sync checked items.’ So if I uncheck it and sync - it’s outta there! 
I delete things from the computer, and when I go to re-sync the iPod, they self-delete.
Now I have song seemingly on the iPod that won’t play. Probably a syncing issue?
Went to Apple store, he told how to resynch: (click off songs, click them back on) but my iPod won’t appear on the iTunes screen. Any help?
Yes. Reboot the iPod. On my old Classic (don’t know if it’s the same on newer models) you do this by holding down the center button, then push an hold the top button while continuing to hold down the center button, until the screen goes black. It’ll take maybe 30 seconds to reboot until the normal screen comes back. Then try synching again.