I would split this up. The first comma's wrong, the others just create a run-on sentence that doesn't change much in meaning when you splice it up into two grammatically correct sentences. I also feel like the moral could be communicated more clearly in the beginning, as "did everything apart" seems too abstract.

"There once was a world with lots of different people of lots of different types who liked lots of different stuff. They played different games, talked of lots of different things, and generally tried to do only what they wanted to do."

I would make it: "But of all those people, there was just one Kail." In creating a list you're making an expectation, the "but" lets the reader know that the expectation created in the list has been usurped.

This is fine.

There should be a comma after "said."

This seems like an odd shift forward in time. Parents have abandoned their children for work, yet things go on? I would say this should happen on the same day. Try, "Kail and the other kids played all kinds of games, but the smaller kids kept losing to the bigger kids."

So Kail, who was a smaller kid, said, "We little kids don't like losing to you bigger kids! If you leave us alone, we'll leave you alone, and we'll all be happy!"

Again, there's a temporal disrupt. The smallest children have been abandoned by their parents and mean baby sitters. All is not well. Try, "Kail and the other small kids wanted to play. But some of them wanted to run and jump, and some of them wanted to do puzzles and sit."

(Wow, is this chic xenophobic or what? Anyway. This is the third time this trope has appeared, so it's time to vary it a little.) So Kail said, "You can play with puzzles and sit while we go run and jump and skip!"

So they each went to where the others weren't, and some small kids ran and played while some small kids puzzled and stayed.]

Temporal dissonance. No! Kail and the other small running kids played tag, but Kail didn't want to talk about the same things as them. (Just shortening things up a bit.)

Make sure you add the comma after "Said."

Kail's a dude? Oops.

You could emphasize "wasn't" here.

I'm curious to see what the pics will be.

Need that comma after "kids" to denote speech.

Don't forget that comma. Also, "do puzzles and sit" is very close to "do puzzles and " so you might want to make it "sit and do puzzles."

Need that comma. I'm not sure this part makes literal sense. You mean sometimes the little kids will play like big kids? If they could play like big kids why were they losing all the time? Maybe sometimes about fairness could go in here.

" And they each went to where the others were, and both big kids and small talked and ran and jumped and did puzzles and sat together. They were all kids!" (shortening it a bit)

Comma after "said."