Is it normal for math grad students to be required to take summer classes to work?
Having observed several places in several different decades: summer funding is always weird and maybe non-existent or not-a-good-deal.

During the 9-month academic year, in the U.S., yes, indeed, people working as TAs/RAs have to be registered as students (because as other people said, these are student jobs), and this entails fees. Most often, tuition is included in the "benefits", but not invariably. 

At every university I've been, grad students registered in the 9-month academic year did not have to explicitly register, nor pay tuition, in the summer, in order to work as summer TAs/RAs. This part is quite anomalous, and looks like a way for some layer of the institution to suck money out of people.

My experience indicates that there's never enough summer support to go around, which effectively creates a labor surplus, which, predictably, leads to certain abuses. For that matter, the usual double-time rate of summer classes is pretty nasty, and not fun to see. A bit taxing to teach, but almost uniformly near-disastrous for the students, who cannot simply "decide" to assimilate things twice as fast.

In some cases, the funding and flow of money for summer teaching/classes is different from the 9-month year, so it is not necessarily deliberately abusive, though possibly irresponsible or negligent, of your dept to not tell you your net pay. If it's really just 600 dollars, it's a ridiculous situation, yes.

But the net pay should be substantial, not next-to-nothing. That aspect strikes me as odd.

In summary, the situation you describe is unusual in that the net pay is so low. Having to sign up for courses and pay summer fees (or, worse, tuition) is unusual. The vagueness is less atypical, since sometimes control of summer classes is not in the hands of the department.