There is no examples in the Superliga, but all other playable divisions have teams with this issue. An example could be HB Køge. They have their rivalry Næstved marked as "local", but their 3 other rivalries with Brøndby, FCK and Roskilde is unspecified. I can list all teams with the issue if necessary.
Scouts and physios, both chief/head and normal. Of course it might also be that their existing staff should only be part time, I don't know. i see many other Superliga teams with part time scouts/physios in game, so maybe that's the norm?
I've tested with SønderjyskE as well and even though they have a full time head physio and scout in game, they're unable to offer full time when offering contracts.