1. The leaf/dust cloud is them digging for Treasures. The shoe, among other things, is one such item. I've had them find flowers, screws, acorns, etc. They'll do this anywhere outside, you'll see it often if you kill time in the Park. You'll know if they fart, it makes a fart noise and releases a little brown poot cloud.
2. I haven't seen that, so I'm not sure.
3. "Pero pero" is them doing lick-lick. Generally in the comics it's done to encourage/heal other yu's that are hurt/dirty (though I'm not sure it heals other yu's in the game). Parents do it to babies also, in order to clean them.