Facebook app not working
Did you sync your iPod Touch with iTunes recently? An iOS device won't download newer versions of an app if they can no longer run on your current iOS version, but iTunes doesn't stick to that rule. So if the Facebook app was updated in iTunes, and then you synced with your iPod Touch, it may have been updated to a version that it can't run, hence the immediate quitting.

You can verify this by going to Settings → General → Usage and tap on the app under the storage setting. You'll see a version number, and you can Google that to see if you have a version that supports old iOS or not.