My expectation is that if you do the things you say -- take grad level courses, maintain high grades, do well on the math subject GRE and even the Putnam, do some research as an undergrad -- that you would be accepted into the grad program of any of the top schools.

The comment by dncheist is well-taken. It is true that grad schools generally do not give you credit for grad courses taken at another school, whether those courses were taken as an undergraduate or not. But that is a small issue. A PhD program is not about taking courses. Most PhD programs in the US have some sort of exam or set of exams that you take early on, say by the end your second year. Some call them preliminary exams, some call them qualifying exams, but the intent is usually to demonstrate mastery of several sets of standard material: algebra, real and/or complex analysis, PDEs, topology, ... the list varies by school. And along with these exams, most schools have a corresponding suite of course sequences to help you master this material. If as an undergrad you have mastered one or more of these subject areas, then you can probably take the corresponding test without having to take the corresponding course sequence. The credit does not matter. And then you can get on to the important stuff -- research -- that much sooner.