You need to focus on the fact that no matter how much optimizing or tweaking you do, a Monk is simply not great at straight-up combat. Monks are good at doing crazy stuff. Monks should not be slugging it out rolling boring attacks, they should be looking for non-standard actions.
A good monk will have lots of ranks in things like Jump and Tumble, and can use the environment to do unusual things both in and out of combat. Thus, try to engineer combat environments that have things to interact with, and encourage the Monk player to use them.
You can take this a step further by using concepts from more narrative games like Spirit of the Century, whereby you encourage the player to suggest the existence of interesting options. A player should be asking things like, "is there a chandelier I could swing off?", and your default answer should be "yes", unless there's a specific reason to say no.
Finally, give magic items which actually work with unusual combat techniques. That means: forget about boosting attack, AC, saves, etc, and focus on cool Wondrous Items that can be used creatively. Boots of Springing and Striding make their already good movement and Jump abilities even better, and if you give a Monk an Immovable Rod and they can't think of something interesting to do with it, then the player should not be playing a Monk.
Incidentally, Monk's best pure combat options are non-standard as well. A strong optimized Monk uses abilities like Trip, Grapple, and Use Magic Device (UMD is feasible for utility / buffs because the DC for wands is only 20 and you can retry fails without expending charges). They really don't need house ruled full BAB or armour or anything because they're not Fighters and shouldn't be played the same.
For example, Trip is a Touch Attack, so you're a LOT less likely to miss against most targets. And since Monks get Improved Trip for free at level 6, you get a free attack to try to deal damage if the Trip succeeds, PLUS an AoO when they stand up. Add Enlarge Person for Size bonus and extra Strength, and Trip becomes way better than just attacking. Monks are better general-purpose Trippers than Fighters because a Fighter has to use a specific Trip-compatible weapon, whereas the Monk can just use their unarmed strikes.
You could say to your players, "Y'know, in this campaign, not all problems can be solved by punching them. And some problems that could be solved by punching could be better solved by not punching, or by punching with strategy instead of with no real plan. What I mean to say is, punching things in the head is a good solution to a lot of problems, but it's not universal. That's the kind of campaign I'm running, here. I probably should have mentioned that earlier."
This method of conveying your point has the advantages of being A) accurate, B) clear, and C) Not taking long.
In my experience, the best way to tell your players about out-of-character and meta-game expectations is to tell them. It's a better way to avoid misunderstandings than subtle hints and in-game encouragement.
Best Answer
The purpose of the questions in Dread is to build a picture of the character which will indirectly answer the larger questions of what keeps the character going when things get tough. As host, you should first make sure players understand this larger premise, and buy into it. Each answer should ideally provide some piece of the puzzle, some insight into what will keep a character motivated when a situation turns grim. Especially important to consider is this bit from the section "Filling Out a Questionnaire" on page 21 of the core rules:
If a player rejects the premise of every question, you might have to have a more serious discussion. But if he's only done this on one question, you have three options.
The first and easiest option is to just let it slide. The rules suggest not every question needs an answer, and if the player's other answers have given you what you need, just move on. If the player has considered it and this is his answer, move on. If the player has a block and can't answer this one, move on.
The key here is to move on. Keep things going.
The second option is one I would not recommend, but I include it because it is an option. Remind the player of the reason for the questions, and insist the premise be honored. Each questionnaire is unique, and the mere presence of the question is a declaration. Yes, your character did have a pet, and yes, that pet died. It's right there on your character sheet. Again, I don't recommend this option, but it's still an option.
The last option is my favorite. Ask the player to honor the spirit of the question while allowing him to reject the premise. This is where that rule I quoted comes in. Push for the "why." "You didn't have a pet? Why?" This option is my favorite because it permits the player a degree of control while still pushing for details that will help build a larger picture.
You mention the concept of "Yes, and..." It's a great concept. But remember, you can also make use of this concept. You can also say "Yes, and..." to any answer you think could have a bit more detail. As host, it's your job to help the players build their characters. "Yes, and..." is a powerful tool for helping them.
Hope this helps.