Auto Pan does that for you. It is located under the map.
Auto Pan: Checking this box will make the map follow the vehicle and thus keep the vehicle in the center of the screen.
Yes. Mission Planner keeps the telemetry log as *.tlog inside logs folder, AFAIK it was in the Documents/Mission Planner/logs. You can play them in MP.
Then, Telemetry Logs > Load Log > Play.