You can take part in all the available competitions at the same time or just focus on one of them. To switch between competitions, use the tabs in the competition window.
Your success points can be counted in for several hours as long as you play with a stable Internet connection. If after a few hours your points have not been counted, then first of all you need to check the quality of the Internet connection, and also try to use the forced synchronization button, which can be found in your profile settings right inside the game.