For the serve, try the view with the camera behind the server. Your forehand shows a nice trajectory for the ball as it travels away from the camera. For the serves put the camera high on the fence and see if the ball traveling away shows for the serve too.
Nothing seems in good focus. Except maybe the camera is focusing on the net ribbon T? Putting you in the center and the net ribbon off to the side might help the auto focus (if central) put you in focus. It might be your camera as you mentioned? But your backhand video seems the best. Why? Check the lens surface and clean if necessary. Shoot in bright sunlight. You can always tell when looking at a video if it was shot in bright sunlight because sharp shadows are cast. If a cloud covers the sun the shadows disappear or become less sharp.
The motion blur with this lighting is excessive. You can't see the racket approaching the ball (stop around impact).
Overcast lighting levels vary a lot. Late in day also the lighting drops a lot even in direct sun light. Try bright direct sunlight to see how fast a shutter the camera's automatic exposure control will select, it might help reduce the motion blur a lot.