Do professors work through all the homework problems they assign to
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.