I disagree slightly with the Affordance Is King point. You should identify the hot spots on the screen, but you should also reward childish exploration. I think most of the screen should be hot. Pretty much everything should be a target, and they should all be big. "Animals For Toddlers" (Eiswuxe) does this well (and is just plan excellent all around, by the way).

I also disagree with the advice that arrows are best for pagination; I think he's overgeneralizing. My toddler is great at swipes, and still doesn't use arrows well. Broader testing may be a good idea here, and implementing multiple paradigms might be the answer.

The bit about not putting the menu on the bottom of the screen is good, though. The kids apps I have on my (Android) tablet often disable the soft button menu and implement their own elsewhere. Of particular note is the pull-down-from-a-collapsed-bar-at-the-top pattern, which my son seems to know how to not get into. Or at least get out of.

And that bit about ads and in-app payments is right on the money. I've bought a few apps for him after seeing he likes the free version, but the ones that result in him launching dating sites get uninstalled fast.