I suppose that you are:
I go in through the doggy door
But out through the cat flap.
Air, in form of wind(moving air), can enter and exit throught a doggy door and exit through a cat flap.
At times, I am cruel to the poor
Cold air, and strong wind can be very cruel to the poor, espetially, the ones who do begging on streets.
But you'll never catch me in your traps.
You can't really catch and trap air. At least you can't trap all of the air
At other times I am kind to all
And provide much needed relief.
Air in form of win can provide relief when it's really hot. Any wind makes it a bit better in these times. Also there is a saying
let me catch my breath, when you are tired and did something really fast. In this case air also provides relief
I am most obvious when everything falls
And I am the painter in chief
At autumn there is usually strong wind (At least where I live), and tree leaves almost all fall to the ground.
Also autumn is really beautifull at times. All those falling
Also, the most obvios one is when you fall, or something falls air is always there.
I think i figured out a bit more
But in spite of all
When your games roll call
Sometimes one of you will shoot me.
I thint there is a game called Paperball, for which you have to use the strength of your lungs, to provide the necessary force to shoot the paper ball. And using breath means using air.
The last lines are the most hard to figure out for me, but i'll take a try
In times, when you are really irritated, or really worried about passing some exam, your breathing can too fast, too slow, you can hold your breath for long. And to pass the test with ease, you deeply breathe in and breathe out several times and you are good to go.