I really don't think I'd recommend living ON campus if you or your spouse are faculty/staff. There are about 40,000 students at the U, and they get pretty lively; it would not be a good place for your youngsters.
However, there are a number of really nice neighborhoods with lots of good rental housing very close to campus, and many, many folks live in those areas without cars. There are parts of the city where it's hard to get to a bus stop, but the campus area is not among them; there are buses through there constantly, and you can get most anywhere you need to quickly and easily. Many folks bike as well, and you'll see bike racks on nearly every corner.
For a neighborhood to explore, I'd strongly suggest the area around Old University, Kendall and Walnut. There are students there, to be sure, but most of them are grad students, hence at least a bit more grown up. There are also plenty of families there, of all ages, and you'd either be right on or within a block or two of a bus line.
In that region, a good deal of the rental housing stock is going to be 50 to 80 years old, which isn't all a bad thing; folks built better back then, and there are a lot of really sturdy and really beautiful places in the area. You'll also find some newer places, some less than five years old, so if you prefer to live in newer buildings, you can find it there as well.
It's actually fairly common in the areas near the university for people to rent apartments sight-unseen, if they can't manage a trip to Madison to scout things out in person. You should try to get the rental company to send you photos of the interior of the unit as well as floor plans, so you can get an idea of the feel of the place. Do check with
Tenant Resource Center about any landlord before you sign a lease; some of them are better than others, and the helpful folks at the resource center can tell you whether you've got a good one or not.
As to your other questions, we moved here when our kids were highschool age, so I unfortunately cannot comment on the elementary schools in the district. I know that most are very well thought of, but couldn't tell you more than that.
I strongly recommend that you check out the "variety of useful links" thread that's stickied at the top of the Madison forum, also. A number of people have added some very good information there for newcomers to the city, and you'd do well to browse the various links a bit.
Good luck to you and your family, and let us know if you develop any more questions as you consider a move. We'll do our best to answer!