Speaking of sports, do Theodosian Elves have any sport that is unique to them ? Just curious.
Chess is something that every Theodosian culture has. I would not be surprised that Elvhen chess use Elven gods instead of Kings, Queens, etc. The Elvhen version of the Knight would the Fen'Harel because of its ability to move through opponent's lines.
Wicked Grace, Chanson d'Argent and Dead Man's Tricks are Human games.
Nug racing, arena duels / The Proving and Diamondback are Dwarven games, though Humans race nugs and play Diamonback too.
That's a good question. I never would have thought of it that way.
I'm sure Thedas humans have a lot of the same sports and hobbies as real medieval humans did; hawking and hunting for human nobles, and gruesome blood sports like bear-bating and ****-fighting for human peasants. (I'd like to think city elves wouldn't do the same with their backyard animals, but being elven doesn't automatically make them all better people...)
I'll bet bull-fighting is a thing in Antiva (or else Orlais; they're flamboyant enough for it) the same way it is in real-world Spain.
I'll bet public executions and jousting tournaments are popular for humans and city elves and surface dwarves too, just like how arena fighting and Provings are popular in Orzammar.
But the Elves don't see to have any games of their own. Probably they spend their time frolicking, hugging trees, riding hart / halla and that's it. No time for games.
I doubt they have time. 
City elves probably work too hard for too many long hours to enjoy too many pastimes. They probably just enjoy cards and drinks at local taverns to blow off some steam between long shifts. (Although, like I said, some unemployed vagrants might enjoy making their stray alienage dogs or backyard roosters fight each other for sport. Blood sports were really popular in the middle ages.)
I can't really think of what the Dalish would enjoy doing; maybe halla racing, but halla are sacred animals, so they wouldn't do that. Since they have to spend all their time hunting for food, crafting weapons and armor to defend themselves, and simply surviving, I can't imagine they'd have much time to do much else besides simple pleasures, like drinking around a fire or exploring fun places in new areas.