What is the earliest possible turn to activate each quest?

Solution 1:

I am just talking about the standard format. And I cheated a bit :D Sometimes your opponent needs to help you out a bit ;)

Druid Turn 5 complete, turn 6 playing Barnabus

Quest: Summon 5 minions with 5 or more Attack.

As first Player (no coin):

1. turn: Quest
2. turn: 1x Innervate + 5-Attack 3,4-drop (e.g. Ancient Brewmaster)
3. turn: 5-Attack 3-drop
4. turn: 1x Innervate 2x5-Attack 3-drop
5. turn: 5-Attack 3,4,5-drop (e.g. Verdant Longneck)
6. Play Barnabus

As second Player (coin):

1. turn: Quest + Coin + 1x Innervate + 5-Attack 3-drop (e.g. Elder Longneck)
2. turn: 1x Innervate + 5-Attack 3,4-drop (e.g. Ancient Brewmaster)
3. turn: 5-Attack 3-drop
4. turn: 5-Attack 3,4-drop
5. turn: 5-Attack 3,4,5-drop (e.g. Verdant Longneck)
6. Play Barnabus

   ---

Hunter

Quest: Play seven 1-Cost minions. Turn 4 complete, turn 5 Queen Carnassa

With or without coin:

1. turn: Quest
2. turn: 2x 1-drop (Jeweled Macaw) (getting one drops or drawing them)
3. turn: Stampede + 2x 1-drop (getting 1-drops or drawing them)
4. turn: 3x 1-drop
5. turn: Queen Carnassa

   ---

Mage With coin turn 2 complete, turn 4 Time Warp
edit: Thanks to scenia, you can of course complete the quest already in turn 2

Quest: Cast 6 spells that didn't start in your deck.

As first Player (no coin):

1. turn: Quest
2. turn: 6x Primordial Glyph) into Primoridal Glyph
3. turn: wait, Opponent plays Arcane Golem
4. turn: Timewarp

As second Player (coin):

1. turn: Quest
2. turn: 5x Primordial Glyph into Primoridal Glyph
3. turn: wait
4. turn: Coin + Time Warp

   ---

Paladin Turn 4 complete, turn 5 Galvadon

Quest: Cast 6 spells on your minions.

With or without coin:

1. turn: Quest
2. turn: Stonetusk Boar + Blessing of Wisdom (attack with Boar)
3. turn: 3x 1-Mana Spell
4. turn: 2x 1-Mana Spell
5. Galvadon

Priest Turn 3 complete, turn 4 Amara

Quest: Summon 7 Deathrattle minions.

As first Player (no coin):

1. turn: Quest
2. turn: 2x Crystalline Oracle (both into Innervate) + Opponent plays Naturalize
3. turn: Innervate + 5x 1-drops
4. turn: Innervate + Amara

As second Player (coin):

1. turn: Quest + Coin + 1-drop (Crystalline Oracle into Innervate)
2. turn: 2x 1-drop (Crystalline Oracle into 1-drop)
3. turn: 3x 1-drop
4. turn: 1x 1-drop + Innervate + Amara

   ---

Rogue Turn 1 complete, turn 1 Crystal Core

Quest: Play four minions with the same name.

As first Player (no coin):

1. turn: Quest
2. turn: Swashburglar (into Shadowstep) + 3x Shadowstep + 3x Swashburglar + 1x Counterfeit Coin + 1x Preparation + 1x Crystal Core

As second Player (coin):

1. turn: Quest + Coin + Swashburglar (into Shadowstep) + 3x Shadowstep + 3x Swashburglar + 2x Counterfeit Coin + Preparation + Crystal Core

   ---

Shaman Turn 4 complete, turn 5 Megafin

Quest: Summon 10 Murlocs.

With or without coin:

1. turn: Quest
2. turn: 1x Primalfin Totem + 1x Murloc from Totem
3. turn: 3x 1-drop Murloc + 1x Murloc from Totem
4. turn: 1x Call in the Finishers + 1x Murloc from Totem
5. turn: 1x Megafin

   ---

Warlock Turn 4 complete, turn 5 Nether Portal

Quest: Discard 6 cards.

As first Player (no coin):

1. turn: Quest
2. turn: 1x Darkshire Librarian discards Clutchmother Zavas and gets killed
3. turn: 1x Darkshire Librarian discards Clutchmother Zavas + 1x Soulfire discards Clutchmother Zavas, Opponennt plays Arcane Golem
4. turn: 1x Soulfire, 2x Succubus every time Clutchmother Zavas discarded
5. turn: 1x Nether Portal

edit: without Arcane Golem played by your Opponent in turn 3 or 4 you can't complete the Quest on turn 5, thanks to scenia again

As second Player (coin):

1. turn: Quest
2. turn: 1x Darkshire Librarian discards Clutchmother Zavas + Coin + 1x Soulfire discards Clutchmother Zavas
3. turn: 1x Darkshire Librarian discards Clutchmother Zavas + 1x Soulfire discards Clutchmother Zavas
4. turn: 1x Lakkari Felhound
5. turn: 1x Nether Portal

Warrior Turn 4 complete, turn 5 Sulfuras

Quest: Play 7 Taunt minions.

As first Player (no coin):

1. turn: Quest
2. turn: 2x Shieldbearer
3. turn: 2x Goldshire Footman + 1x I know a guy into Shieldbearer (Opponent plays Arcane Golem, otherwise not enough Mana to complete Quest)
4. turn: 1x I know a guy into Shieldbearer + 2x Shieldbearer + 1x Frostwolf Grunt
5. turn: 1x Sulfuras

As second Player (coin):

1. turn: Quest + Coin + 1x Shieldbearer
2. turn: 1x Shieldbearer + 1x Goldshire Footman
3. turn: 1x Goldshire Footman + 1x Frostwolf Grunt
4. turn: 1x Frostwolf Grunt + 1x I know a guy into Shieldbearer + 1x Shieldbearer
5. turn: 1x Sulfuras

Good luck! :P