Did you already try to reset your ipod by holding your sleep and home button until the Apple logo comes back again? You will not lose data doing this, but it can cure some glitches.
Usually apps not meant for your iOS version can't be installed on your device. I tried this on my ipod, you can select the apps for a higher iOS, but they won't install.
If resetting does not help, set it up "as new device" without using the latest backup, in case you would be reinstalling damaged files. How to set up your iPhone or iPod touch as a new device
(I'm embarrassed that I didn't think to try resetting it myself.) I tried it and, yes, it's functioning much better. It's still sometimes a bit slow, for example, when I ask it to find someone in Contacts. I have a LOT of apps to test, but so far none has crashed while trying to load! :-) *Thanks very much.* You were a much bigger, faster and easier help than either the Genius Bar or Apple's telephone support.