Scenario 1:
In most cases, mulligan for your combo!
Since your deck has no Wild Pyromancer and very few early drops, I believe your best chance for taking control of the field is your Auchenai Soulpriest and Circle of Healing combination.
Grabbing control of the board against aggressive decks is key to winning when playing as the Control Priest. Therefore, this combination is so important, that you should mulligan aggressively for it.
That said, if you have a 1 and 2-drop (Northshire Cleric and Acidic Swamp Ooze) or a 2 and 3-drop in your hand in addition to the Circle of Healing, playing those will give you enough board presence early on to protect your health until you do draw your Soulpriest (or at least have enough chance to draw it as you would with a mulligan).
Scenario 2:
Mulligan Thoughtsteal. Below is an explanation why.
I am assuming that Dark Cultist or Injured Blademaster is a better turn three play than Thoughtsteal in most circumstances, so you would rather have one of those in your hand. Since you have two of each Thoughtsteal, Injured Blademaster, and Dark Cultist, you have six cards that you could draw by turn three that would be as good or better than Thoughtsteal.
- Mulligan
- You keep 2 cards and mulligan Thoughtsteal.
- There are 5 cards as good or better in your deck.
- Your deck has 27 cards.
- Chance that you draw something worse for turn three: 22 / 27
- First turn
- You drew something worse.
- You have three cards in your hand and you draw one.
- There are 6 cards as good or better in your deck.
- Your deck has 27 cards.
- Chance that you draw something worse for turn three: 21 / 27
- Second turn
- You drew something worse.
- You draw another card out of your remaining 26 cards.
- There are 6 cards as good or better in your deck.
- Chance that you draw something worse for turn three: 20 / 26
- Third turn
- You drew something worse.
- You draw another card out of your remaining 25 cards.
- There are 6 cards as good or better in your deck.
- Chance that you draw something worse for turn three: 19 / 25
The chances of all of this happening in sequence is (22/27) * (21/27) * (20/26) * (19/25) = 37%
That means you have a 63% chance to draw a replacement Thoughtsteal or one of your Dark Cultists or Injured Blademasters by your third turn. Therefore, mulligan that Thoughtsteal!
Remember, in a control match-up, you have a very good chance to draw it again later, when it is more useful.
Best Answer
I don't believe that you'll find a non-legendary that would give you the same kind of reach and flexibility as an Archmage Antonidas. There simply is no other card in the game that gives you a fireball spell in hand.
The fireball that it gives you when you cast a spell is much more powerful than any other card you may get. If it lasts more than a round then the effect snowballs out of control. By that time in the game (turn 8-10) you could easily dish out 17 (2x Fireball + Archmage Antonidas attack) to 20 points of direct damage (2x Fireball + Archmage Antonidas attack + Frostbolt).
The only sort of viable alternatives that I've been able to find is a Ragnaros. It's a 8/8 on the board and deals 8 damage at the end of your turn, albeit randomly. If there is no answer for the Ragnaros, it's helping deal out significant damage per turn. Although he's vulnerable to Big Game Hunter.
Ysera might be another viable alternative because of all the cards it can possibly give you. It's a bit more of a defensive alternative where as Ragnaros is more aggressive.
Edit: After reviewing the cards, Jeeves may not be a bad idea either. Since you'll be lacking cards and most likely playing almost all your cards every turn in a mech mage style deck Jeeves can help you replenish your hand fairly quickly and you might be able to draw the extra few cards that you need.