Taking you with me achievement
I have tried to get a couple of achievements through the starter deck duel events. I tried to get the "Taking you with me" achievement (where you have to win on the same turn you sacrifice 6 or more permenants) by casting Lich-Knights' Conquest in the golgari snack time deck. I have done this achievement twice. Both times the opponent did not surrender so I know the spell resolved and I killed them on the same turn. At first I thought the achievements might not be connected to the stater deck duel event, but I noticed I made progress toward the lifegain achievement with the same golgari deck through food.

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Sire of Toast#99907 commented
Starter Deck Duel games don’t qualify for ANY “0/1” achievements but they do count towards “over any number of games” achievements. It’s not explicitly stated for some reason but only draft and constructed “matchmade games” qualify. I think this is the same with Midweek Magic events.