Does the video recording stop automatically when the MicroSD card runs out of space?

No, not if Loop Recording is turned on. When the MicroSD card is full, the system will automatically replace the oldest unlocked video files with new ones.

Full Answer:
If Loop Recording is turned on, the monitor will continue recording without interruption when the MicroSD card becomes full.
Loop Recording saves video in smaller clips. Once the card runs out of space, the oldest unlocked video file will be automatically overwritten by the newest video file.
You can turn Loop Recording on or off in Monitor Settings > Loop Record.
If Loop Recording is turned off, the monitor will stop saving new videos once the MicroSD card is full. To free up space, you can format the card by going to System Settings > Format MicroSD.