It generally means the battery has a too low voltage for the Flic to operate. Did you try a new battery?

It's obviously a bug in either the OS or the Bluetooth chip in the phone. Can you post a hci snoop log that records from the phone boots until it doesn't work?

You can pair the button to multiple devices and assign the same config. But the button itself can only be connected to one device at a time. So if you have two paired devices nearby and the button has gone to sleep, when you then press the button, it will connect to the device that responds quickest.