You only need internet if you are using cloud functionality and remote monitoring.
Also, the Astrobox should automatically re-attempt a connection to the AstroPrint Cloud in the event of a connectivity interruption.
You should be able to use your AstroBox locally without internet though
Nope. The GCODE file is completely sent to the printer before a print starts.
Therefore, the loss of internet will NOT stop a print in progress.
The only thing that will stop a print in progress is a local power failure to the printer