Are daily-quests player or day specific?

Solution 1:

It's randomly selected for each player. Today my daily quest was "Summon 20 minions that cost 5 or more" that I rerolled, getting a "Win 5 matches with Rogue or Warrior". As you can see none of them match the one you got.

Regarding the high number of opponents of the same class it mostly depends on the game mode you're playing, but can also be influenced by the time of the day when you're playing. In Arena you'll find a lot of Paladins, Mages and Rogues, the classes that are currently stronger in that game mode, while on Constructed you'll see many Druids, Warriors and Paladins, that are very strong in the current meta. Also from my experience I've seen that late night play promotes slower decks (I get more Control Warriors or Dragon Priests), while on lunch break or during the day more generally I get faster decks (usually Paladins).

Solution 2:

The Hearthstone Gamepedia wiki mentions that the Class Combination quests are randomly selected, and my (and my roommates) experiences seem to corroborate this.