The easiest way to explain these interactions is that Lord Jaraxxus is summoned as a minion, and then his battlecry takes effect, changing him into a hero. In this way he isn't affected by cards like Demonfire or Shadow Word: Pain but is still affected by those secrets (Mirror Entity, Snipe, Repentance and Eye for an Eye).
Interestingly, he is still affected by Sacrificial Pact, as he is a Demon even as a hero.
That doesn't explain anything at all - it's exactly what I said. His Battlecry changes him to a Hero, so he should never be affected by Mirror Entity, Snipe, and Repentance, which all activate after Battlecries, and have Minion in their card text, not Character.
He is still affected by Sacrificial Pact because his type is Hero, but also retains the race of Demon
Sorry, looking at my explanation I realise I didn't include the most important part - cards don't 'fizzle' in Hearthstone.
Of course, you've already had a great explanation, but that's what I was trying to say - all effects are stacked, and then they play out. So because he is a minion, his battlecry effect is put on the stack, then the secret effect is stacked, and then the interactions play out.
My explanation was not very good, but I was trying o give a short, readable response XD Sorry again.