I'd like to think I'm more Holistic than most guys, so I can try and give it a shot. At the very least, I can try to tie in some of my favorite Holistic characters and show how their thought processes affect their problem-solving style.
I think of Holistic Problem-Solvers as sort of like tinkers. Remember, they don't see the problem as "if-then;" instead, they see this swirling mass of influences resolving into the current state of affairs. So to tease out the solution, they play with one dial and see what happens. Maybe they want to convince some friends of theirs to help with some light physical labor. Instead of, say, using logic to convince them to help or paying them to help, they turn on the charm and say something like, "Oh, if only I had some big strong men to help with this..." Maybe this works, maybe it doesn't. But whatever reaction there is, it's something the Holistic person can adapt to. You figure out their personality quirks, and then you poke and prod at them a little. And when it works, the Linear folks are shocked. "How did you... but... you didn't even... huh?!" (As another example, take everyone's favorite rogue, Tom Sawyer. How does he convince his friends to do his fence-painting for him? By loudly yelling, "Man, painting the fence sure is fun! I'm sure glad I get to do it and none of you do. Pfft, there's nothing I'd rather be doing right now than painting this fence." See also: potatoes.)
This doesn't always work, especially when the Linear thinkers are on to the game. There's this scene in House of Cards, where Holistic thinker Frank Underwood is trying to manipulate someone into doing his bidding. He asks for something totally trivial, completely meaningless to the other person. The other person says something to the effect of, "Wow, I can't believe you're trying to manipulate me." They'd figured out his M.O., so even if they couldn't see the immediate cause-and-effect, they knew he was trying for something.
Regarding Lord of the Rings, I think a Holistic thinker would talk about all of the different parts. They'd emphasize the part where Aragorn leads the charge against Mordor, not to immediately get the Ring from point A to point B, but to distract Sauron and his forces from the two little hobbits sneaking around the back way. Rather than say, "It all comes down to destroying the Ring," they'd focus on how it's a team effort, and how defeating the Witch-King and routing Saruman and inspiring the Ents to action are all necessary parts to defeating the threat of Sauron forever. But ultimately, I think they'd have to fall back and accept that Lord of the Rings is a very Linear story, and its plot ultimately does come down to getting the Ring from the Shire to the Mountain. They might, perhaps, be more interested in a story like A Song of Ice and Fire, which is much more about managing political relationships and manipulating each little piece until the right person can take the Iron Throne. (Though I haven't been following ASoIaF, so I could be wrong on that.)
Regarding the difference between men and women, you've finally given me the perfect opportunity to share this video with everyone! (If you're Linear, here's the Holistic answer to this video: perhaps she already knows the nail can't come out of her head. The goal of her complaining is not removing the nail, but pain reduction.) Regarding feelings, I think you could say the difference is that for Linear thinkers, feelings are something you resolve, whereas for Holistic thinkers, feelings are something you manage. Like, let's say your SO does something that makes you mad. The Linear thinker thinks, "I'm mad about this! I want to stop being mad about this. How do I go about doing that?" Whereas the Holistic thinker thinks, "This has added 'mad' to my emotional system. Now I have to modify everything else in the system to balance it out." Or they might think, "My SO did this thing that made me mad, but it's not as simple as saying, 'Don't do X.' They only did it because of A, B, and C personality trait, so if I want them to stop doing X, I have to change A, B, and C."
Or for another example, the cliche of the girlfriend always saying, "I don't know where I want to go to eat; why don't you pick?" Remember, it's not about the nail. The real thought in the girlfriend's mind is, "Does my SO know me well enough to predict what I want to eat?" The girlfriend picks up all of these things, because to her Holistic mind, that's what managing a relationship is like. When it's her turn to pick where to go, she puts all of that to mind. "I'm hungry for X, but I need to watch my calories, so Y might be better, but SO prefers Z, but we'd have to get dressed up for that, and I haven't shaved my legs, so he'd probably like Q, since it's pretty similar, and restaurant V has a low-carb menu..." A Linear thinker hates this! If he could hear this thought process, he'd say, "What's that? You're hungry for X! Great! Let's go get X!" But it's not about the nail!
Regarding the grumpy kids, a sidebar. There was this scientist who wanted to find out what things affected a judge's ability to make fair rulings. So they polled one area for several days, checking a certain kind of crime and logging the number of times the judge chose to pardon or convict the defendant. They found a disturbing phenomenon: at the beginning of the day, the judge would pardon about 40% of the time, but that number would decrease steadily throughout the day, hitting 0% moments before noon. Then, after lunch, it'd jump right back up again. That's right--the judge was making more convictions because he was hungry. If you were to ask the judge whether he was completely fair all day, he'd say, "Yes, I judge strictly based off of the facts." But the numbers tell a different story; the judge was actually trusting his gut--literally!
Holistic thinkers see through the veil (or at least, they believe they see through the veil). If a person is happy or sad or angry or grumpy, it's caused by a huge mish-mash of everything in their environment, not just one dominating factor. If a traveling salesperson makes a sale, it's not just because of the value of their product, but a whole host of factors, down to the crispness of their suit and the wideness of their smile. Holistic thinkers are great with vague, "soft skill" concepts, like "mood" or "charisma," because they don't mind having to spin all the plates at once. They can't tell you what makes a charismatic person charismatic any more than a Linear person can, but they can see a lot of things that make someone more or less charismatic, and they can affect those things.
So I think part of this 3.B. question is a hidden question: "How do I manage interactions like this?" (See, Holistic thinking right there!) If a Holistic person in your life comes to you and asks, "What's got you so grumpy?", try taking a step back for a moment. Your pat, top-of-the-head answer may not be the whole answer. For example, maybe you're usually really accepting and cheerful when the kids are playing, but for some reason, it's setting off a nerve in your brain today. There might be something more going on. It could be something like you said--"I'm stressed," "I'm bummed about the latest movie reviews"--or it could be something even more intricate. You were standing in line for a long time for burgers today, and when you came home, SO was frying up ground beef, and the smell reminded you of that feeling of frustration and impatience you felt back then.
On the other hand, look at this from the other angle. You came home, and when the kids came up and started playing, you snapped at them and told them to play by themselves. Your grumpiness is throwing the home atmosphere out of whack. It's not about the nail; the question isn't, "How can I help you stop being grumpy," but "How can I get your grumpiness out of my network?" So maybe the correct answer to, "Why are you so grumpy?" is, "I'm sorry, I'm going to find something cheerful to counteract my previous grumpiness." So say something like, "I was grumpy because I hadn't gotten the chance to see how lovely your hair looks this evening."
(I was thinking about your aside, "his wife is dealing with it just fine." Because she's been in the environment so long, she's figured out how to balance everything else out to make it less of a strain on her mood. ...Or maybe not, and if they get any more rambunctious, she'll snap and not fully know why. That's always the big weakness of Holistic thinkers: they often try to handle problems indirectly, either through Being--changing their own environment and diluting the negative effect--or through Doing--subtly manipulating the other person until the problem goes away from the back end. "Manipulating" is such a bad word for it; when it works, you don't even notice it. Shoot, you thank them for it! But when it doesn't work, it feels very... well, manipulative.)
[Oh, and this is why it's so difficult for me to explain this. Because I always have to explain it two ways: through Being, and through Doing.]