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The New Horizons 3.0 update is here and we're hosting an eShop gift card giveaway! Each participant will also earn up to 60 forum bells. To participate, visit the event thread in the New Horizons board here. Good luck!
There's a chance they could. They usually don't mention them until much later, although, I don't know if they'd do weekly things for Animal Crossing like they did with Fire Emblem and Kid Icarus.
As for what is usually given out, if anything, just various goodies and merchandise. Some events are bigger than others (Like B/W where they had part of Rockefeller Plaza reserved and a bunch of tents and stuff), but they do some small ones too (Zelda Hyrule Historia--they gave out these cute little sword bookmarks).
They have also done events where they have actually sold the games early too so...