Yes, this works both ways. However, you always have to create an account with Logi because the data is stored/transmitted via the cloud. There is a delay of about 3-4 seconds.

Yes, the camera can be switched on and off via app or web browser. Independent of one's own internet access.
It is also possible to switch on private mode. Then the camera continues to run, stream (to watch with the app) and also record to the cloud storage, but there are no push notifications. So it's helpful when you have a child's birthday party and people are constantly walking through the door.