Rikard,
1) The lights shouldn't flash while in the cradle. Flashing lights indicate a problem. They should be on (charging) or off (fully charged). The screen should be responsive and usable while in the cradle. The exception would be if the battery is drained and the lights aren't on solid.
2) The Pro has been reported to turn itself on (very briefly) at midnight according to it's clock. Anything else is definitely not normal.
3) Again and indication of a problem.
You might try reflashing the firmware again. If that doesn't help, I would assume the remote (and maybe the charger) is defective.
Jim L