Honestly, I haven't tried it, but my USG was already without power, my controller rarely runs (I don't have a CK yet), and I haven't had any problems yet. What does "losing the config" mean for you? You won't have to re-adopt the thing...

Yes, the settings are basically stored on the individual devices - no matter whether gateway, switch or access point - and for the pure operation of the devices, the Unify controller is not necessary. For monitoring, updates etc., the controller must of course be running, as everything is managed via the controller. I have the USG, a 24 port switch, 2 access points and the Cloud Key as my setup.