yes of course, it puts your mind to work ; of course it helps you especially in times when you have to make a difficult decision .... it is good to train mentally
I'm sure about it! I can even notice that when I am not active, I feel lazy, distracted, tired and exhausted. And when I exercise regularly (I often follow the https://betterme.world/articles/ suggestions) and keep the right sleeping schedule, I'm full of energy and therefore my brain works better.
It is unlikely that physical activity will help win the tournament. But it will help to improve your health. And this is much more important than winning the game.