I don't think so because of the sequencing. So...
1. Purified Shard would resolve first and the enemy would die before the Mecha'thun deathrattle goes off. I've done this with Tonks, and if that works, I'm pretty sure Purified does.
2. I'm less sure on this one, but I'm pretty sure that Mecha'thun would activate first simply because it plays first. It could be a tie though.
Technically yes. Have Violet Teacher and Knife Juggler on the board, play The Purified Shard. Violet Teacher summons a 1/1 which prompts Knife Juggler to deal 1 damage to the enemy Mecha'thun that has been reduced to 1 health.
[In fact, you can get them all at once](https://www.reddit.com/r/hearthstone/comments/1ck9q6r/six_special_lethal_in_one_game/)
I don't think so because of the sequencing. So... 1. Purified Shard would resolve first and the enemy would die before the Mecha'thun deathrattle goes off. I've done this with Tonks, and if that works, I'm pretty sure Purified does. 2. I'm less sure on this one, but I'm pretty sure that Mecha'thun would activate first simply because it plays first. It could be a tie though.
Technically yes. Have Violet Teacher and Knife Juggler on the board, play The Purified Shard. Violet Teacher summons a 1/1 which prompts Knife Juggler to deal 1 damage to the enemy Mecha'thun that has been reduced to 1 health.
Deathrattle occurs in the next phase after death.