I'm wondering if this is case especially for time-consuming work like programming problems in computer science spanning several thousand lines of code.
Or when a professor teaches a new course they are not intimately familiar with e.g. did not take it during undergrad.
For professors that don't do this, how (if at all possible) do they address homework concerns during office hours? Is it recommended for professors to know the ins and outs of all problems they assign?
For many professors, undergraduate courses are simple enough that when they go over 1-2 problems, they can grasp them quickly. For others, they go over solution manuals and previous notes to recall and remember how to do them. Some assign office hours to TAs, where the TA deals with students and their homework. Keep in mind that many schools do tell the professors which courses they will teach next semester/year, so they have plenty of time to prepare and develop their notes and slides.
I would be more worried about professors teaching graduate level courses that are not familiar with. It seems that many issues/inconsistencies occur at this level.
Short answer:
Yes and No
All your cases do apply.
Dutiful professors actually work problems before they are assigned as homework problems to students. Such instructors may even go through the trouble to frame their own problems for homework.
There are many who provide a set of problems from standard solutions and web references. They may not have worked out all of them, but rather a small subset of them.
There are, however, some who are fairly (or entirely) new to the subject and are experimenting with questions that are available as exercises in the syllabus with the students. They actually learn along with the students as they teach the topics progressively.
An instructor, in general, would be actually a mix of the above. There may be some topics that are the same as the previous syllabus that the instructor knows well enough to frame own questions, and some others where she/he might prefer to use textbook exercises. There may as well be a few other topics which are new in the updated syllabus which she/he may wish to experiment on.
I teach at a small all-undergrad department. No TAs here, so yeah, if I'm not using some kind of on-line homework system (which I have only ever done in service classes, and I'm trying to give it up) then I have to do all the problems at some point to produce a key.
Now, that doesn't mean that I've worked them out in fine detail, nor that I have done them before I assign them. In my upper division classes I just chose a few that look like they go to my learning goals and might be interesting. If several students complain that problem X is too hard I look again and then maybe post some hints to the LMS or give them a less ambitious goal for the problem; or maybe not: tackling the occasional hard problem is a skill that they should be exercising.
As for how I help students when I haven't done the homework first myself, well, mostly it's a cakewalk. For one thing I never work the problem for the students: I just keep asking them related questions and suggesting concepts they might try to apply. For another, I've been doing this subject for longer than a "traditional" student (i.e. 18-24ish years old with little real-world experience between the end of secondary school and the start of college) has been alive. I choose problems on the basis of the concepts they exercise and I can usually tell that at a cursory reading, which means I can decide on a strategy (or several strategies) that will work when the student shows me the problem that has them stuck.
Do professors work through all the homework problems they assign to students?
No, not all the time. Dealing with your secondary issues in reverse order:
Re: Recommend knowing all the ins and outs? Perhaps in an ideal world, but I would recommend prioritizing other things first. When prepping a new class, my protocol looks like: (1) Folders, (2) Schedule, (3) Syllabus, (4) Tests, (5) Homework, (6) Slideshows, (7) Handouts. (In addition, there are clerical duties like: export PDFs, set up Blackboard, submit duplications to office services, make attendance sheet, make backups, etc.) Keep in mind that sometimes we are assigned a course only a few weeks, or even a few days, beforehand. In those situations an instructor would not have any time to do more than select a list of promising-looking exercises on the topic from the book. In later semesters perhaps I will go back and work individual exercises to be completely comprehensive, and refine the assignments (but likely some colleagues would consider that obsessive, and think the time better spent on service or research, etc.).
Re: How do they address concerns? The thing is, the whole point to real knowledge is to have an abstracted higher-level framework in which the minute details fade to unimportance. I can look at an exercise set and say, "these are all really the same piece of work". This is frequently the case if you look at a text explanation written next to a series of mathematical manipulations; the line-by-line English descriptions may be totally identical for a given exercise set (even if, say, the numbers and variables are all different). In this sense, a professor working out the details of individual exercises may be entirely redundant, whereas for a student they're constructing a new perspective on the topic (what initially looks different gives way to awareness of commonality). That said, it may be good to check and see if there are details or complications unfamiliar to students, so that can at least be mentioned ahead of time in class. But: If one truly understands the subject in general, then any exercise or detail should be easily addressed on the fly.
Re: Professor not taking undergraduate course? It should be pretty unlikely for a professor to be teaching a course they didn't take as an undergraduate; undergraduate curricula are usually pretty standardized, and a PhD holder has taken all those classes and much more.
A lot of professors do solve their problems first. Among other things, it's always a good idea to check the problem to make sure it's solvable with the techniques the students know so you don't have egg on your face later.
Many people teach the same class over and over, it's common to prepare the problem once at the beginning and then reuse them. They can create extra problems and rotate them each semester to avoid students memorizing solutions from past courses.
Relatedly, common textbooks often have teacher's only problem banks. The answer book isn't really that hard to get, but many students don't bother.
Some have TAs handle the questions. Since it's tedious, boring work, it's great for getting the TA to do it. Often TAs solve the problems in problem sessions anyway.
Some profs just don't care and give problems without checking. When someone complains after they discover an impossible problem (eg. angles of a triangle add up to 190 degrees) they say "oops" and carry on. Students get upset but when's the last time a professor got fired for making a mistake in his homework problems?
So your answer is: Yes, some do.
However, it sounds like you're just shocked at how long you took with your assignment, so are assuming it can be done faster. It's always possible that there's a faster way to do it than you don't know. Also, repeatedly doing the same problems quickly makes you very efficient at it: Even students can, just by solving problems throughout a single course, get good enough to get an A. Imagine how good the professor must be: They are probably smarter than the average student, know more techniques and methodology, and have taught the class multiple times.
In most courses I was a Teaching Assistant in, the answer was: "No, the Professors let the Teaching Assistants do that, and they report back any issues they find." (And sometimes we would handle coming up with and checking HW assignment problems all on our own - when there were several TAs in the same course, anyway.)
I'm not saying this just to relate my personal experience - it is literally something a Professor can be assisted with. If you are a Professor and you want to give homework, try to pressure your department to assign you a part-time TA - even if only for this work specifically.
This will vary tremendously based on the professor - I'm not sure you'll get any answer other than "sometimes".
In my own experience...the answer is sometimes.
I've had professors where it's very clear they've only had their TAs do their assignments, or where the level of testing hasn't even gotten that far. On the other hand, a professor I worked with used to sit down every week for a meeting with his TAs, work through the problems, suggest extensions that the group presenting the homework that week could do, etc. That was a tremendous amount of effort, and likely wouldn't be repeated in later years once the problem sets were fine-tuned.
This might be just my experience, but I've been now almost 4 years at a UK university, and all (with one exception) of my main lecturers would know their coursework by heart. Furthermore, for most of the math classes I took, the lecturers would actually work through the coursework problems after all coursework has been submitted/returned.
And for project-like coursework, e.g. writing a not-so-small computer program, or performing such-and-such experiment, many of the course leaders would maintain either a thick booklet with a lot of guidelines/examples/marking schemes/etc, or libraries of work done by students from previous years. And these materials are usually passed to the new course leader, when course leaders change.
In summary -- yes, in my experience at a UK university, professors work through all the coursework problems they assign to students.
