I've seen this often in anime, and I've sort of guessed what it means. But I want to know what it means exactly. So does anyone know what the Eye-Tongue thing means?

As far as I know they mostly do that to provoke someone или to just make someone angry. Maybe they also do it to Показать someone that they uhm..how to say..give a shit about the other person they are Показ it to.
(sorry for my english)

Well, I'm not sure about other countries, but in the United States, Ты know how little kids will stick their tongue out at another kid as a rude gesture? Well if Ты know what I'm talking about, it's the same thing, the gesture is just a little different. It's just a slight cultural difference.

Another example off the вверх of my head is the gesture to call someone to come to you. In Japan, they call people over with their palm facing away from their body, where as in the United States, they have their palm facing towards their body. It's similar to that.

A gesture of disrespect towards someone in a childish way...kind of like "NANANANANA, COME AND GET ME YA OLD GEEZER!" или something...

Pretty sure that's just a gesture to get one riled up/angry it's kind of like saying "come and get me",don't think it's much еще than that and Ты mean like this right? :P