[RPG] What ability should be used when resisting Intimidation

Caveat: the answer to this question is going to depend heavily on your DM, and how they feel about and portray the use of social skills. Some DMs don't much care, and will allow you to use these skills with little penalty or roleplay required. Other DMs may view social skills as potentially game-breaking, or at least very important to handle correctly. If you intend to use your Intimidate often, I would recommend finding out your DM's policy on social skills, and adjusting your expectations accordingly.

That said, in general, Intimidate is meant to be a representation of your character's ability to bully or scare someone. So, to address your three points:

1. To "cow a target into taking some other action" generally means to frighten them into doing something other than what they had planned. For example, let's say you're fighting a group of goblins. One of them runs at you with his sword. You roll an Intimidate check, and describe the action as "I tower over him, my staff smoking from my last spell and my eyes glowing with power". A successful check might cause the goblin to attack someone else instead. Alternately, if you are particularly successful (let's say you rolled high on your Intimidate check, and in the previous round you critted a different goblin) you might scare him into losing an action or even running away.

2. Can you use Intimidate on NPCs? Yes, but remember: it's a representation of your ability to bully or scare people. Intimidating townsfolk might get you the information you need about the evil lich lord - but it might also cause the NPC's bigger, more powerful friend to come after you for bullying his pal. Likewise, bullying a shopkeeper once might mean he hides his best wares next time you're around, or simply closes up shop entirely to avoid you.

3. (2a) You didn't ask specifically, but it's the flip side to #2 so I wanted to address it. As a DM, I don't allow social skills like Intimidate or Diplomacy to work on other PCs unless you both a) roleplay it out, and b) have a very good reason for doing so. Skill checks are meant as an abstraction; when you are interacting strictly with other players, you don't need that same level of abstraction. It's very frustrating, as a player, to be forced to act a certain way because you rolled badly and your fellow player rolled well. But again, this depends on your DM and your group, and their take on social skills in the game.

4. Do targets become hostile out of combat, and what does it mean? They will very likely become hostile - as I mentioned, even if your initial target is too scared to fight back, his big bad friend might not be. Or, if you Intimidate a member of the criminal underworld, say, he might act on his fear by drawing a weapon on you, forcing you to fight (and spend resources) when you might otherwise have not needed to. And 'hostile' doesn't always have to mean 'immediately attacking'. A hostile bar wench, for example, might get revenge by poisoning your drink, or by spreading vicious rumors that bring the city guard down on you. A hostile merchant might raise his prices, or avoid selling to you. And so on.

Intimidate, like the other social skills, can be very powerful when used correctly. The key is to ask yourself whether bullying or scaring someone is the best way to get what you want out of a situation. If the penalties are worth the payoff, then go for it.

There are optional Morale rules that can be found on page 273 of the DMG.

To determine whether a creature or group of creatures flees, make a DC 10 Wisdom saving throw for the creature or the group's leader [...] On a failed save, the affected creature or group flees by the most expeditious route.