No, the 12V battery, the hybrid battery and the combustion engine all need to work for the vehicle to function.
I also think this is correct. Apart from the fact that any reasonably modern cars will usually go into limp-home modes, or even stop driving, if errors are detected, I seem to remember something about an electric drive motor needing to spin backwards when driving using the combustion engine. That feels wrong as I write it, but I remember it being something fundamental to the planetary "Power Split Device".
Depends what, but mostly yes. If the battery fails then the engine is used permanently. If the engine fails then the battery can be used for its limited range.