I also forgot to mention, there can be quite large side forces on a door frame when a heavy front door slams shut, etc.; that is probably why the lightweight concrete has cracked around the old plugs. Get a door damper for the door, and it will be gentler.
Then it might not be so relevant anymore, but to answer your original question; yes, lightweight concrete screws are screwed directly into lightweight concrete without pre-drilling. However, it should be pre-drilled with a smaller hole in lightweight expanded clay aggregate/leca.
