Will the Task or Event Update Its Status Automatically?

Yes — but only from monday.com to Google Tasks. Status changes made in monday.com will sync to Google automatically.

For Google Tasks to monday.com, updates are not automatic.
You’ll need to click the Sync from Google button on the account authorization screen each time you want to pull those changes.

For Google Calendar events, the sync is fully automatic and works both ways — updates made on either side will reflect in the other in real time.