Something that clearly violates the physics of the setting e.g. "I jump to the moon", sure. I get that.
That is one case, yes. Pretty much the only one that I would invoke, though it could also be 'genre logic', like "no you cannot make Dynamite in this D&D game, even if you can describe the necessary steps to do so, and carry them out in character." That can usually be colored as 'physically impossible in this world' too of course.
But it's also within a DM's purview (or should be!) to just say no to an attempt to climb a wall that's simply beyond the ability of the character, even if the character thinks she can do it. (though if it came down to it I'd have the player roll anyway, and hope she rolled a 20 so I could get the point across that nope, you ain't climbing this one)
I honestly am not disputing this, in the sense that from a certain kind of play, this is true. I mean, its really the point
@Ovinomancer often makes when he talks about how 5e is generally run. In the kind of play that I prefer, there simply isn't such a thing as an 'unclimbable wall' that will probably come up in play. It is quite possible that nobody CAN actually climb a given wall, but that supposition should be tested and could be found untrue. Different story games actually have various approaches to this, and they certainly don't all use a 'say yes or roll' approach either. A game might, for example, demand certain resources be expended, or that a PC display certain traits before they can act in a given way. I mean, fictional position is a thing in most games. However, a 'wall' in the terms I play represents some sort of fiction that could be overcome to move towards a character goal, and it exists not to channel play in a designated direction, but to be an obstacle of some sort. Canonically in some games, like DW, an 'unclimable wall' could exist, but only so as to provoke the players into making moves that assert how they face that obstacle. Its like the canonical dragon who cannot be hurt by Hack and Slash (its scales are invulnerable to mere swords and such). The PC maneuvers himself into provoking a bite from the creature and then stabs it in the mouth! This is clearly quite dangerous and says something about this character. Likewise the wall might be unclimable until the character accepts help from his rival, or expends something precious, etc.
Correct, it's not important most of the time; but as you're the one who brought the example up you're stuck with it now.
The bigger play here wouldn't be the specifics of getting the robe for free, it'd be the general idea of putting one over on the game.
I think it is everyone's game, there's nobody to put one over on. If the wizard wants to have a free robe, chances are the other participants in the game are going to see that dimly. At best there's a very pissed merchant who's likely to denounce him at an inopportune moment and claim he was 'bewitched' or something. Mostly I just don't see that kind of toxic behavior much from players.
Their characters ARE their interests. As players, they have one job. One. Count 'em. And that one job is this: advocate for your character.
I'm not so sure about that. I think the players are interested in the quality of the game. I found this to be very true when running my 4e games. The players had a lot of ways to influence the game, and I never saw any of them take advantage of that. At most they might advocate for leniency, perhaps, but then there would be a discussion "well, you did run out far in front of the party, that was bold, but it got you ganked." or something like that.
And like it or not, advocating for one's character includes trying to squeeze out whatever advantages, be they in-game or mechanical, one can for it (and a player not doing this is abrogating the responsibility of advocating for the character to the fullest means possible); and it's the DM's job in turn to keep this in check.
Well, we just play differently

I don't keep anyone 'in check'. That certainly isn't my personal goal or method.
Thing is, "being the players' ally" is the sort of thinking that very quickly leads to fudged rolls and other non-neutral refereeing.
I don't have this problem either. We are all playing the game and the point of it being a game is to 'play to see what happens', there's no logic that would lead to fudged rolls or any sort of bad faith. OTOH I would rather produce an interesting outcome in the end than just be hard and fast with "the way things are." We fought a combat in my last HoML game, and the PCs got crushed. The player's dice were abysmal, and on top of that the base DCs probably need to be tweaked a bit. I think they SHOULD have been able to win, but I'm not wiping out the party because I had some monster jump them and their dice were cold. Call it what you like, but now they have a fun mystery to solve, why are they still alive? I don't think that's the greatest most perfect way for the game to go, maybe it even undermines playing to see what happens a little bit, but the point is to have fun in the end. This becomes especially clear when you're running a game you wrote, its like "everything I did was because I wanted to do it, there's no arbitrary game rules here to blame it all on!"
And that right there is part of the campaign's interest: self-sustainability. The DM has to run with an eye to keeping the game going, one week to the next and one year to the next; while the players can happily ignore any such considerations if they want to.
Meh, in my experience the logistics of play rarely lead to a specific game continuing for more than a few years. Nor does anything I do seem to undermine my campaigns particularly. I am just going to have fun now, and if the story that comes out of it was interesting to play, people DO keep playing. Nor IMHO is it my job to tell the players what the content of the game needs to be, or know what is 'best' for anyone.
Which is, in a broad sense, one place where I run aground on the story-game concept: it's hard to have a mystery, or secrets in/about the setting, if there's little or no hidden fiction; and a large part of play often revolves around discovering secrets and-or solving mysteries.
Well, in my HoML game that I'm running now, there is at least one BIG mystery, which is pretty nebulously drawn at this point, but will presumably occupy one of the PC's minds over time "what happened to the Bear Knights of old?" I have ideas which I think will play out fairly well, but technically I am not holding back some secret fictional explanation. There's also a more immediate mystery about why the PCs are alive. That totally arose randomly out of play, I sure didn't set it up! I don't even know what the answer to that question is, frankly (though I think it will probably tie in with the other mystery. OTOH it could be related to the other PC perhaps? Maybe he's got some kind of a guardian or something. If so there will no doubt be some sort of cost to that in the future...).
Everything.
Time, and the careful management thereof by the DM, is perhaps the single most important element in making a game world believable and inhabitable.
Well, so Gary said. I never paid too much attention to that, frankly. I mean, I guess in 1979 I pretty much accepted it at face value, though even then I recall being rather dubious about it. I mean, yes, you can run a game like that, time could be a resource of that sort. OTOH even Gygax didn't necessarily run all these games in strict linear time order. I'd even bet that there was a retcon or a flashback or two in there. Regardless, I don't have to run Gary's campaign, mine is a bit different and it can handle a flashback here or there. I don't really do it very often anyway.
Well, in my mind a DM who has to retcon to that extent just shouldn't be a DM. Retcons are bad bad bad bad bad.
Luckily there isn't a licensing authority for GMs. People keep coming back. Nor frankly do I remember the last time I did any sort of retcon myself. The most I've done is provide a bit of a post-hoc explanation for something in combat, like when the Warlord pushed the orcs. When the fighter reacted to that my description was that the Warlord gave him some advice or an order that provoked the orcs to change direction. So even though their minis technically occupied a certain square at a given time in the turn order, FICTIONALLY what the Warlord did on his turn, which came next, modifed that, they actually moved to a different place, one he chose.
Why?
Because a retcon invalidates whatever aspects of previous play that are being retconned and-or overwritten, meaning that going through said play in the first place was a complete waste of time for everyone involved.
I am still asking why? If they had fun, and if it lead to a determination that there should be some other different fiction it doesn't seem pointless at all. Certainly these sorts of categorical qualifications of things don't work well for me.
Put another way, the existence of retcons means that even once fiction has been established, well, it still hasn't really been established because it can always be retconned later. Bleah.
I mean, OK, I'm certainly not telling you to have different preferences. As I said before, I have not found a retcon to actually be necessary, not in long enough that I cannot really recall when it was.
Sigh...
It's nothing to do with children and everything to do with advocating for the character to the fullest of your ability and squeezing whatever advantages you can, which includes pushing the rules envelope until it pushes back.
Well, I think that players DO identify with their characters. However, when the action in the game is dramatic and interesting, and leads to fun outcomes, what more can people ask for? Yep, the Dragonborn Sorcerer was slimed by bullywugs and he kicked the bucket. Everyone thought that was pretty amusing, as he was played as being super picky about his appearance. Maybe the player would, in theory, keep the character around. OTOH she got to play a Pixie Wizard instead, and that character was a lot of fun.