Do professors work through all the homework problems they assign to students?

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.

1. 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.

2. 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.

3. 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.