It's in the rules:
You will be randomly matched up with one player from each of the other 3 Elements.
I know, but are you matched up with all 3 simultaneously or 1 at a time? That's what I'm trying to clarify, sorry if I didn't make that clear.
How about: "The 1st Avatar" ... then the next one would be: "The 2nd Avatar"
Each generation has it's own Avatar.
Yes that's good, but in general linking the icon to the name of the Event is preferred. This way if someone sees a cool icon and mouses over it they can clearly identify which event awards it. While naming the award "#th Avatar" might seem crystal clear, there are people who won't connect that with The Avatar as an event (trust me it happens). Think of the icons as advertisements for future editions of the event so including the event name in their description is important.
Expanding it to 12 separate elements would completely change the event. Becoming an Avatar is about mastering the 4 core elements, but in the case of Elements-Avatar, you really have to master all 12 elements. As such, you have to use all 12 elements to win.
I believe this event has a nice blend of EtG battling mixed with the Avatar theme.
The main issue is that in EtG there really isn't anything like 4 Core Elements. Each one is meant to be treated equally. If anything, the "4 Core" EtG elements would probably be Entropy, Fire, Aether, and something else that is used most (Earth maybe?). As far as having to master all 12 elements, the current set of rules doesn't guarantee this. A player could go the entire event never being given certain elements as Marks or Elements. They can also choose to never use some elements at all. As more and more Marks/Elements become available it could be argued that the need to master more elements becomes less and less since you increase the chances you'll get the same elements as before letting you use the same decks as before. The TV series/movie addressed only the classical Greek foundation of 4 elements that were the basis for everything in existence (for example Earth + Fire = Life in the Greek model). EtG obviously adds new elements to this classical equation (Aether, Entropy, Time for example). I don't see how taking the original framework and building even more onto it changes the entire event. You still have to master elements and do everything, just now there are more elements to master overall that just 4. Being an Avatar might be about mastering the 4 core classical Greek elements, but shouldn't being an Elements Avatar be about mastering the 12 EtG Elements?