I want to know if all players receive the same daily-quest or if this is random for every player on the same day. For example, today i got the quest "Win 5 Battles with Hunter or Druid". Did all players got the same quest today, or did they get a random quest out of the quest-pool? I am asking this, because on some days i get a lot of opponents with the same class, I was wondering if this is because of an overall daily-quest. But when dailys are different for every players each day this would be just bad-luck...
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).
The Hearthstone Gamepedia wiki mentions that the Class Combination quests are randomly selected, and my (and my roommates) experiences seem to corroborate this.