OK, inexperienced DW GM here, but I'm going to give my stab at zmook's 1-4, partly to see if experienced GMs agree with my calls.

1.  It's the thief's turn.  He successfully trips an ogre who was about to bash the fighter.  What bonus, if any, do you give to the fighter's next attack against the sprawling ogre?

For all of these, note that I don't really think there's a clear 'turn'. When zmook says this, I'm going to assume that he means that, on table consensus it seems like it's the thief's chance to say what he does.

My call on this one depends somewhat on my description of the ogre. If I've described it as lumbering and ungainly (likely, given my mental image of an ogre), I'm likely to just tell the fighter to deal his damage, no move required. I might make him Defy some Danger (with either Con or Dex, most likely) to avoid the ogre thrashing about trying to get up, but that depends a lot on how I've fictionally positioned it.

2.  The ogre, back on his feet, is winding up with his stone mallet to club the fighter into next week.  The ranger acts, and hacks the ogre from his exposed side.  What bonus, if any, do you give to the ranger's attack?

Here it depends on what the Ranger is trying to accomplish. Is she trying to outright kill the ogre? If so, it's probably just deal damage, but if that doesn't drop the Ogre right away (and it might well not), that monstrous swing is going to happen -- the threat was ignored, so it's a golden opportunity, right? -- and the fighter will probably take some damage and get to roll Defy Danger (Con) to avoid some real problems, given that I spent some description on the rippling muscle's as the ogre wind's up and the dried blood coating the mallet head.... If she's trying to distract the Ogre from actually taking that swing, she's probably making a Defend (the fighter) move, which might incidentally get to inflict his damage if she takes that option on a hit.

3.  It's the bard's turn.  He rolls Defy Danger to disarm the ogre, and the brutal mallet goes flying into the darkness.  Now the fighter Hacks the disarmed ogre.  What bonus, if any, do you give to the attack?

So I'm not sure why it has become the bard's 'turn' here. I obviously gave the Ranger a chance to respond already, right? I needed to say what happens before, which means probably resolving that threat I prepared here. But suppose the ranger didn't exist, and the bard jumped in to try to deal with the fighter's problem, and I resolved the bard's action as you described, with the mallet flying away and landing with a heavy thunk on the ground behind the ogre. Now what I do for the fighter all depends on the move I made after that. Is the ogre abandoning his weapon and now trying to just crush with his fists? Pulling out another weapon? Turning to try to recover the mallet? If the first two, the fighter is probably just hacking and slashing normally, but the ogre's weapon presumably doesn't have the 'messy' tag and it probably makes the ogre's other moves less effective. I might even reduce the damage the ogre does in return.  If I think the ogre is getting the mallet back, I'd probably say, "Sure, Fighter, you've got a clean shot to deal your damage, but it'll pick that wicked thing back up if you do. Or you could Defend the mallet from being picked up, right, if you want to keep that ogre away from it instead."

4.  The cleric drops a weighted, entangling net over the ogre.  It comes around to the fighter's turn again.  The ogre is now disarmed, netted, and the wizard just blasted it for big damage.  What bonus, if any, do you give to the fighter's attack?

"It's helpless. Do you want to kill it outright or capture it to haul back to town? I mean, (Bard), I'm sure you know someone who would love to have a live captive ogre to experiment on, right? Who would that be?"

While I agree that the motivation is very important, I really don't see that laying out the core quests in a constrained form like this helps us get there, precisely because they are so generic. 'Discover the truth of Blackrock Tower' doesn't tell us any more on its own than 'Seek the Holy Grail'. Even 'Slay Horlath the Ogre, a blight on the land' is only slightly better in that regard.... why are you going after Horlath instead of the other blights? (There are always plenty of blights upon the land. Many of them can't be slain, of course....)

In practice, I think a Paladin is going to see something in the fiction he wants to take up. Then he's going to look at the goals under Quest and see which of the Quest goals is best fit. Sometimes that's going to be a fairly tortured reading, as the answers you gave demonstrate. Worst case scenario, he looks at the quests and thinks that none of them really fit the mission he wants to take on, so he can't use the Quest move. Best case scenario, one seems to be a pretty good fit... but then what have you gained by finding one and stating it in those terms? It doesn't define the task more clearly, or provide the motivation (presumably the Paladin already had motivation for taking this on). Maybe it shades your perception of the true nature of the task a little, but when you're stretching 'Defend ...' the way I see you doing above, I don't think this is really important. It certainly doesn't help me see when the Quest is complete, which I think is part of my problem with some 'Defend ...' uses.

The more I think about it the more I think it should be as simple as: "Describe the selfless and heroic task you have set yourself." (Rather than expanding the list.) That matches my conception of what a Paladin's Quest conceptually should be -- a selfless and heroic task -- without needing to be fit into any prewritten boxes.

Somewhat related to this is the question of how long a typical Quest should last. I would think it should be something like usually two to three sessions, not infrequently one or less, and only rarely much longer than that. I think you want the Quest changing with some frequency so the GM can tailor the vow to what is interesting in the current environment, and not be stuck with what he told the paladin 6 months ago. This consideration is what is leading me to prefer Quests that have pretty well-defined ending conditions.

Okay, I love the Paladin's Quest. It's dripping with flavor, and it drives the paladin to action, which is great. Basically it seems to me that the paladin in the group should very nearly always be on a Quest for something. 

But if that interpretation is right, there are several classic questy things that don't seem to be easily expressed with the three fill-in-the-blank quests that are listed. The ones that have occurred to me-- there may be more -- are:
Rescue ______ from ______'s captivity.
Bring ______ to ______ that it may be destroyed.
Retrieve _____ and return it to its rightful owner.
Find the fabled lost artifact ______.

I mean, you can fill in many of these blanks with values from classic stories, right? If it's supposed to be easy to go on a quest, I would think these or some similar ones should be added. I think I'm going to do that, unless some of you good people convince me that he existing list is adequate or that Paladins should have a little more trouble establishing quests all the time...

On a related subject, it seems to me that the defend blank from iniquities quest might not necessarily drive action from the paladin as much as the others would. This is purely a perception, not from actual play, though.  Anyone with experience using that style quest have any comments on how it worked out?