Is it inappropriate to ask department staff for syllabi previously used for courses I am to teach in upcoming year?

In the one department I have taught in, it was stated that you have to run the course along similar lines to previous/other instructors (some courses run w/ multiple sections so they need consistency). So absolutely, ask for the existing syllabi. And good luck with your first teaches!

You should also check to see what classes have your class as a pre-requisite. You want to ensure your students are prepared for what follows.

A possibility to consider in addition to the other answers: there may be a syllabus template, and you may be required to use it. So yes, ask for existing documents.

I do the adjunct hiring for my community college department, and we have a certain set of problems that we see repeatedly. One is people teaching a course at the wrong intellectual level (usually too low). Another is not assigning enough work or not giving enough feedback. I would not want a newly hired person to go to an administrative assistant, get a syllabus from one of those people, and then replicate that.

I suggest that you start by talking to the person who hired you and who will be making the decision on whether to rehire you later. Say that you're interested in seeing what people have done in the past when they did a good job of teaching this course. If you said that to me, I would respond by handing you a sample syllabus and also telling you in writing about my expectations. (Well, actually, I would have done those things without your having to ask.)

It is completely appropriate, and can be valuable in the context of the greater curriculum. When I used to teach classes, I would ask the previous lecturer for the syllabus. Most importantly, I would also speak to whomever is teaching a follow-on class to see if there's something that should be added or removed based on the changes to the general curriculum.
As an example, I was teaching a sophomore electrical engineering course as an introduction to analog circuits. I ended up spending a week on the actual device physics because students were going to be required to have either device physics or analog design as a follow-on.

The first thing I did when asked to teach already existing courses was to ask the previous lecturers for copies of syllabi, teaching materials, exams, tutorials....

Probably depends a little on office politics, but I was just handed everything in a nice manner and took over from there.

Actually, it is quite appropriate. You could also ask for a list of the professors who previously taught the course. You could then ask them if they are willing to share materials to help bootstrap the course.

In fact, the administrative assistant might not have easy access directly, partly due to the pandemic. But the professors would normally retain their materials.

You might also find recent course information online if each course has a website or if professors have websites with course material.