Great to hear! 
Yes, that should be automatic, although understandably only works if your companion has an internet connection (any chance it wasn’t connected to wifi?)
Directly flashing the SD card is the most robust way of getting a working version, but that’s generally only required if something has gone significantly wrong with the normal update process (via the system page) or for very major updates (e.g. going from normal companion to our new Companion Beta).
Yes, via the system page, although there were some recent issues that mean that’s now only possible with the latest version of Companion (0.0.30)