The pie throwing is just for fun and for a little bit of pero. Some people use it for communications as well, you can put short messages into the thing. If you're short on cash, there's a really simple daily mission for it as well.

Licking them off is just to clean up your card (which is an indication to some people that you're actually active!) and to cash in on the pero reward

No significance to the square that I know of. It also occasionally shows up in the Shards event, but so far I haven't seen any logic behind its appearance.

The selection of cards serves no other purpose than bragging rights, or just showing the cards you like. The only event where a "hand" is used is the Shards PvP event, but it uses it's own "hand of cards".