I know that in 4th edition, you're allowed to make an Intimidate check against a bloodied enemy to send it running. As far as I've seen in the rules, there doesn't seem to be any real limitation other than this. What should I do if a player chooses to max out their Intimidate score to take advantage of this? Should I always allow it, or case by case? What about situations with elite or solo enemies?
First things first: from what I can tell, that rule has never been errata'd, so we can go by what's in the Player's Handbook.
I would say allow it at pretty much any time, but don't forget that you can add circumstantial bonuses and penalties (+2 or -2). If the enemy's side is clearly winning, the bloodied creature would be less likely to retreat. Many monsters won't speak the characters' language, and thus the intimidate check would suffer from a -5 penalty. You can optionally choose to give extra penalties for checks against elites and solos (-2 and -5, respectively, would be appropriate, but I haven't checked those numbers to see if it would make it too difficult).
Also remember that intimidate is a standard action, so a character who is taking his turn to intimidate an enemy can't do much else. If they fail, they also can't try to intimidate that same target again during the encounter, so it's usually a safer bet to beat the enemy into submission rather than trying to intimidate it. Besides, 4E combats can drag sometimes, so having a few monsters surrender when they're already bloodied might just speed up the game.
Finally, be creative with the term "surrender". Some creatures might lay down their arms and cry for mercy, offering anything that might spare their lives. Others might run away, instead, and that means they could come back later with bigger and more powerful friends. Turn a successful intimidate result into an interesting plot point, making your adventure more dynamic.
Yes, always allow the check. That doesn't mean you can't control the success rate though.
The check is vs. the target's Will defence, OR a DC set by the DM.
So if you think it unlikely a given monster would be intimidated then set the DC higher, or if you don't want the monster to be intimidated then simply tell the player he failed after his roll, he doesn't need to know the roll was pointless.
Don't do this all the time though, of course, you must not let the players ever know the role was pointless, let the dice decide when you're not so bothered about the outcome.
Some suggestions for guidelines. Older RPGs concerned with this issues had morale checks at 50% of the original number of creatures and then 25% of the original number of creatures. In the case of the larger beasts (epics in 4e parlance) this would be figured as a factor of hit points. I would set your DCs accordingly. It should be most difficult if the target is bloodied and all his compatriots are otherwise still standing, and the most easy if there only a few left and facing a total kill.
Well, not everything reacts to being intimidated the same way--it doesn't always have to be surrender or retreat--it could just be avoiding attacking the scary guy, or deciding the scary guy is the biggest threat and needs to go down first, or fighting worse and granting combat advantage, or in some cases, well, you know what they say about cornered rats becoming tigers . . .