Was brainstorming for something else and wondered if there was a way to incorporate poker straights ( a series of sequential cards) into Hearthstone. Apologies if this has been done before, and better, but here were my first few ideas. Fair warning, not completely sure the wording is right, but hopefully it's grokkable.
This would be the Wish spell you can get in Dungeon Runs.
I don't know how effective these would be given the requirements.... that being said I like the idea. I've had a different yet similar idea for a spell for sometime that would deal damage sequentially left to right on the board (or right to left). Deal 1 damage to leftmost minion, 2 to the next, 3 to the third, and for as many minions there are on the board.
The card would be called "Cascade - Deal sequential damage across the opponent's board starting at 1 damage"
Perhaps you could choose which side of the board on which to start the Cascade, however this might be a bit difficult to play around.
Not quite like the straights idea, but made me think about this again.
I like your ideas and I would welcome this "mechanic" in the game. This is mainly because how your cards require you to build your deck more differently and you have to have a specific set-up, in order to get the big rewards:
1) Archmage's apprentice - holding onto a 1-, 2- and a 3-mana spell by turn 10 and then casting all of them (with your 4-mana mage) at the same time.
2) Prince Nazjak - this reminds me of Defile in the terms of having a specific board state. However I think, that this is the weakest in comparison to your other ideas - a) kinda hard to execute imho (if your opponent is familiar with your deck strategy, they will try to counter it), b) not to mention, that 5-mana cards have to be really rewarding (+ reliable), in order to be worth the inclusion in an aggro deck. I would suggest to boost the buff a bit - his effect should set the stats of your minions to 6/6, which also goes in line of what you are trying to design. Inb4 kudos for the prince theme.
3) Medivh, the Mastermind - my favourite card from all of your ideas. So I like how the requirements are hard to fulfill (surviving by turn 10+, playing really expensive minions, spending 10 mana for an 8/8 body and not answering your opponent's board state and then 10 mana again for the reward itself), but the reward is really powerful and worth all your efforts. I think, that there would be a competitive control deck running this card, not to mention, that tons of casual ones will form just for the sake of it.