Does the Apple Watch monitor your heart rate all the time or only when you put your finger on the screen

ECG is ONLY when you explicitly take a 30 second ECG reading.

Your pulse is measure every so often in normal use, but not frequently.

When you are exercising, your pulse is checked more frequently, or if you actually bring up the heart rate app.

The watch checks your heart rate about every 10 minutes and constantly if you are in a workout through the workout app.