Can I make it such that when a mod deletes a song from the songlist, it times out the user who requested that song for x amount of time? Also, is there anyway that Nightbot can make it so that only the channel owner can ban people (by unbanning people when mods ban them?
No, it is not possible to make nightbot time out the user who requests a song that was deleted.
No, there is no way to have nightbot enforce timeouts from your mods. I would highly recommend just talking to your moderators and making sure that only you should be doing bans, and they should do timeouts.