The first one is about Hugo Strange, I looked this up, but did not find anything about it specifically.

Professor Hugo Strange, Psycho-Analyst
Arkham Inmates, Character, 1/3, Common
Each of your opponents may not recruit a non-army character with the same name as a character he already controls.
Boost 1: When Professor Hugo Strange comes into play, if you control another Arkham Inmates character, each of your opponents discards a card from his hand.

1. What if you control one Hugo already, the next turn, attempt to boost him for the boost effect, do you control...another inmates character to satisfy his condition?

2.Purple Man , Doom - Character 4/4- Activate KO a resource you control. Move target face-down resource an opponent controls to your resource row. That opponent puts the top card of his deck into his resource row.

Now what if I play this effect on an opponent, targeting a face down resource, but then he responds with using it before hand, what happens? (assuming the play is legal, do I still KO a resource?)

3. Divided We Fall, Plot Twist- Play Divided We Fall only during your attack step.
- You may stun a ready Secret Society character you control with cost greater than that of target character an opponent controls. If you do, ready that character and move it to your front row. At the start of the recovery phase, that character's owner moves it to his front row.

I know I must choose a valid target upon playing the card, but I do not have to do anything else until resolution, correct?
If that is the case, can you ready a stunned character? If so, cards like Press the Attack or otherwise would work?

(1) When you recruit a unique character (as Hugo is), right before they finish coming into play, all characters in play with the same name are put into the KO'd pile.
The boost happens after Hugo comes into play, which means that the other Hugo is gone already, and thus his ability will have no effect.

(2) Note that "Activate" is part of the cost, while "KO a resource you control" is part of the effect. (the arrow seperates them.) If your target is illegal on resolution, then the entire effect is negated. Since the target is a "face-down resource", your opponent having turned it face up will make the target illegal on resolution, negating the entire effect. Which means you don't have to KO a resource. (the cost of activating still remains paid, however.)

(3) Yup, "target character an opponent controls" must be chosen on playing the effect, but nothing else.
You cannot ready a stunned character, as the game rules specifically state that a stunned character cannot ready. However, if you target a stunned character, the rest of the effect will still work. (you'll still move it to your front row, and move it back at the start of the recovery phase.)

Alternatively, if your opponent stuns his character in response to you playing Divded We Fall, the effect says "you MAY stun..", so you can always choose not to stun a character you control, and thus not take control of anything.