Do graduate schools have more courses & less of a student-teacher ratio than undergraduate? On average do grad schools offer more distinct courses than undergraduate schools?

If that is the case, then it must mean there are more professors teaching in graduate schools. And common sense tells me that there are less students in grad school than undergraduate. 

Does that mean there a disparity in the student-teacher ratio in grad and undergrad schools?
I will limit my answer to American universities.

Do you mean more courses per semester, or more distinct courses that are listed in the course catalog and offered with some regularity?  

If you mean the former: in any given semester, most departments offer many more undergraduate courses than graduate courses.  This must be the case for the reason you suggest: the number of students taking undergraduate courses in that department is usually far in excess of the number of students taking graduate courses in that department.  

But most departments offer largely the same undergraduate courses every year (or sometimes, on a two-year cycle).  Many undergraduate courses at a large university have multiple sections running concurrently.  The most popular courses -- e.g. freshman level courses for non-majors -- are in some departments run in ten, twenty or more sections each semester.

If you mean the latter: yes, many -- perhaps most? -- departments with large graduate programs offer a larger number of distinct graduate courses over the years than undergraduate courses.  Each course is populated by a smaller number of graduate students, but graduate students stick around for at least as long as undergraduates and, unlike undergraduate students -- recall that I am talking about American universities -- take the vast majority or the entirety of their courses in a single department.  Sometimes graduate students repeat courses or stop taking courses at a certain point, but in many PhD programs -- including the one at my university -- graduate students are required to take a minimum (positive!) number of courses every semester, some of which are unrepeatable.  

In fact, most departments devote much more thought to which graduate courses they are going to offer in the next year or two years, both in order to make sure to offer enough courses for their students to take, and conversely to ensure that enough of their current students will want to take any given proposed course in order for that course "to make", i.e., to meet minimum enrollment requirements and be run.