Hello John,
For point 2 you're right: sliding the app out of the task manager in iOS will prevent if from running in the background until you open it back.
This is not *our* design but a limitation from iOS (which is understandable).
Technically, with a transport different from UDP, the background task allows Linphone to be woken up about every 600 seconds to refresh the REGISTER on the SIP server. Also, the socket is left open by iOS and will wake the app up when data comes by. It's largely different from Android where the app is running all the time.
As for point 1, when first launched the application should ask you to authorize the use of push notifications. If you didn't see this, I believe either you installed the app over an existing Linphone application, or installing it from iTunes did something unexpected which prevented the pop-up from appearing.