A: Yes, they can! AP controls Theater of Horrors, and one of AP’s opponents has lost life this turn, so AP can play cards exiled with Theater of Horrors, just as the card says. If the Theater was supposed to be limited to only allow playing cards exiled with it that you own, it would have to say that. It doesn’t, so its effect carries no such limitation.

There isn’t a particular rules backing for this, because it’s simply a matter of the card doing exactly what it says it does. In general, cards do (and do only) what they say they do, even if that’s not always what would be most convenient for the player who put them in their deck.
Approved by Callum Milne Cards: Theater of Horror; Eyes Everywhere

A: Yes, AP will be able to see the fourth card down while resolving Brainstorm. There are rules that say if the top card of a player’s library changes during the announcement of a spell or ability, that player don’t reveal it until they’ve completed announcement of that spell or ability (401.5 and 401.6). There’s no such rule if the top card changes during resolution of a spell or ability. In the middle of Brainstorm resolving, after the player has drawn three cards, they can look at the new top card before they have to put two cards back to the resolving Brainstorm.

Approved by Nathan Long Cards: Experimental Frenzy; Brainstorm

A: The Mishra’s Factory will be a 2/2 in each of the cases. Both abilities will set the Factory ability to a specific P/T (layer 7b), so we will be looking at timestamps. Dependency doesn’t apply here – it only applies within a single layer or sublayer. If different parts of an effect need to be applied in different sublayers, dependency (or lack thereof) is determined separately for each of those layers.

As such, a Mishra’s Factory that’s animated while a Humility is already on the battlefield will be a 2/2 right from the start, because the animation effect will have a more recent timestamp and will therefore be applied last. This also means that a Celestial Colonnade animated while a Humility’s out will have flying and vigilance, because the animation effect gives it those abilities after Humility has been applied to it – the same goes for other lands that grant themselves abilities with their animation effect.

Approved by Callum Milne  Cards: Humility; Mishra’s Factory