If you open the run app, the watch uses its own GPS. However, the app does not have a way to wait for it to triangulate, and the route is not recorded. Try to open google maps before opening the running app and wait for google maps to show that it found the satellites, you can see that it is not searching and the circle on the centre is blue, then open the run app. It can take a bit of time to get the right location sometimes.
No according to the documentation. The watch uses the phone's GPS but no the health app.
The GPS antenna in your Watch will be used when using the Samsung Health feature and the health platform apps, even though your Watch is connected to your smartphone.
By the way, it is a hit and miss affair with tracking your run. On the phone the app shows the map and you can see when it is ready. The watch starts the countdown whether location is ready or not. That's why I suggested to open google maps to force the location and then start the running. This should be done when the phone is far enough that it is not connected to watch.
Whenever I run, there is always a day that it does not track the run on the map.