I can't get rid of the red dot notification.

On the watch swipe down for the top to see the notification. One seen, the dot goes away. You can turn off that indicator by go to the Watch app on your phone>MyWatch>Notification turn off the Notifications Indicator slider

There are a few notifications that you will receive on the watch that do not correspond to anything on the iPhone. One particular one that we received last night/this morning on the watch is the Happy New Year notification. That also resulted in fireworks across the watch screen when you tapped on it.

Otherwise, the information that Illass provided is on point.