If you don't mind, I'll deal with them in reverse order:
Play DVD conforms to the Kodi action 'xbmc.playdvd()' - This has the visibility condition 'System.HasMediaDVD' automatically added to it, which (despite the text of the visibility condition itself) will only be True (and therefore show) when Kodi detects a disc - from what I understand any disc - in a cd/dvd/bluray drive attached to the system.
Radio and Live TV have many different specific actions, depending on exactly what they're doing - playing a live channel, displaying a list of channels, displaying recordings and so on. The automatic visibility condition for these is either 'PVR.HasTVChannels' or 'PVR.HasRadioChannels', depending whether is a live TV or radio link. For these to be visible you have to have a Kodi PVR backend enabled and returning some channels - if there is no backend enabled, or if the backend doesn't tell Kodi it has any channels, they won't display. See
PVR (wiki) for details.
(The PVR links can be forced to always show, whether there is an enabled backend with channels or not, from script.skinshortcuts settings - 'Settings'/'Systems' > 'Addons' > 'System' > 'Dependencies' > 'Skin Shortcuts' > 'Configure' > Toggle 'Never hide PVR shortcuts' > 'OK' - however, forcing them to always show doesn't mean they'll actually work if you don't have a PVR backend set up)