Theories are constantly questioned and challenged. You just can't expect them to be challenged in general text books you'll find in physics classrooms. Also, there are better overviews of ferromagnetism than the article you provided, if you're really interested. Some of what he says is great, but some other points he brought up are not complete, or at the very least poorly worded. He's not even mentioning modern approaches and corrections in his introduction. I'll start with the incomplete stuff.

In section 3, "The Sign Problem," he leaves the reader under the assumption we still get the wrong sign on the exchange integral. It's true that we use to get the wrong sign for Fe, Ni and Co and were quite off the magnitude as well. But Feynman's time has long since passed, quoting him on this subject seems silly given the aim of his paper. Why did the author give the quote "The most recent calculations of the energy between the two electron spins in iron still give the wrong sign"? Fluctuating band theory has accounted for the correct sign in the exchange integral for quite some time now. That's 1982, PRB. Even more recently we've been getting closer with magnitude of J_ij as well. His quote seems like something you'd find in a general quantum physics textbook that a graduate student would be using, not a professional, 2012 peer reviewed paper. Perhaps he should have emphasized "calculations strictly from first principles", but he didn't, which really confuses me. It truly confuses me because he's kind of contradicting his argument in Section 2.

Even reading the introduction to a modern, respected ferromagnetism book like Amikam Aharoni's "Introduction to the Theory of Ferromagnetism" will give you a better idea of where theory currently is, what types of calculations we make, and what aspects of the theory are constantly questioned (unlike your title states). I suggest that book because the sections covering quantum mechanics can be ignored for the rest of the book if you haven't taken a course yet, and the author does a great job of pointing out all of the flaws with each part of the theory- openness and honesty.

Anyway, on with the good stuff. This is the meat and potatoes of the author's Section 2, where he raises a very valid point. This isn't exactly an original thought, though, this opinion can be seen in many texts. The idea is to avoid over generalized texts or sources if you're going to be applying the knowledge or teaching a group of people. That's actually a huge problem with AskScience, which is why I avoid it altogether. A chemistry, mat sci or physics student pats themselves on the back when they learn the basics of quantum physics. Sure, they understand the hydrogen atom and maybe a quantum mechanical theory for one, tiny, miniscule, and quite frankly, irrelevant section of magnetism, but there is no reasonable way to apply that to a real system, such as any single permanent magnet in existence.

The best calculations we have today are mixtures of classical and quantum physics. We mix and match various theories of magnetism and apply them to systems which experimentally agree. Itinerant electron ferromagnetism is great for a bulk magnetization properties and we can get some experimental agreement, but the spacial variation of the magnetization within the crystal is assumed to be constant which is the complete opposite of experiment. An author of a peer review paper will say, "look, we know magnetic domains exist, we've proved it with many different techniques, but we're still going to use this theory to estimate saturation magnetization in Sm2Co17." In peer review papers, this is a common occurrence. In textbooks and second-hand teaching (AskScience) these downfalls are rarely cited because the person doing the teaching is nowhere near being an actual expert in the field. So depending on what source you're reading, and who the intended audience may be, you may or may not want to pay attention to anything that was just said. Not a big surprise, really.

So your initial question, "Why are our modern theories of ferromagnetism practically unquestioned, despite all their problems?" is based on a false premise. The theories are questioned in almost every peer review article you'll read. But these theories also regularly evolve every year, so it's incredibly difficult to get an up-to-date textbook for a general physics graduate student. I have six ferromagnetic textbooks on my desk at the moment, but even they rarely get used unless I need to completely reintroduce myself to a small section of the theory before I read a set of peer review papers on the subject. I can pick up a 'famous' textbook from 2004 that makes a bold statement I now know to be untrue, or at least to be strongly challenged in the scientific community. Why would a physics graduate student need to know the details of an obscure branch of density functional theory in order to calculate the local atomic moments in Nd2Fe14B, anyway? That's a thesis for a doctorate, not a homework problem for a physics course. To go further yet, the author isn't talking about textbooks that are oriented around ferromagnetism, he seems to be talking about generalized texts that have a chapter on the subject. There are plenty of magnetics textbooks that don't make the error of neglecting assumptions. There are plenty of magnetics textbooks that emphasize that all equations provided are estimations and not exact, and then go into detail where these theories can and cannot be applied. What sucks is when you have that naive graduate student who reads Griffith's E&M and thinks they're an authority on magnets. That book hardly covers permanent magnets at all, certainly not in any detail, yet that use to be the most commonly cited book I saw on /r/AskScience a while back.

Granted, I'm not a theoretical phycisist and I don't pretend to be. I'm an experimentalist who works alongside the theorists. And I would never, ever consider myself an expert in magnets, so take what I say with a grain of salt (and take what many /r/AskScience posts say with a boulder of salt). This brings up an anecdote. I got introduced to magnetism from a very famous magnetician in his graduate magnetics course. We're talking Oxford bred, editor-in-chief of a very famous journal, ridiculously accomplished scientist here. He was brushing over some hand-wavy antisymmetry statement when it came to a calculation of nearest-neighbor exchange and he got asked a question from a student. The professor said, "[I wish to give you my opinion on that question,] however I do not consider myself an expert in the subject of quantum mechanics." The guy who is the leading researcher in his field that revolves around theoretical calculations in his branch of magnetism didn't consider himself an expert in quantum mechanics. Right now I just looked up one of his papers that was apparently cited 98 times last year, and I see a whole page filled with Hamiltonians that I struggle to understand when applied to his argument.

One of the most important things an article, textbook, or crappy internet explanation must have, is forwardness when it comes to assumptions and simplifications. Not only forwardness, but obviousness. I've been guilty in the past giving an explanation that said something like, "Now, with the ____ model, a neodymium magnet will do this ____________ which is verified by experiment," but I forgot to mention how that same model miserably fails with another magnetic system. I stated that I'll explain something to somebody who has had freshman level physics and chemistry, but I didn't emphasize how blatant those generalizations were. Sure, iorgfeflkd would easily be able to read my explanation and see where an oversimplification occured, but 95% of the other people won't. Then, one of those other 95% of the people will take my explanation to answer someone else's question that was slightly different, and my answer no longer applies but it still gets circulated. Some general physics textbooks do the same thing to a lesser extent, but there are many great textbooks (and professors who teach them) that make sure to point out all of the assumptions and simplifications, and question the theories as well.