If I apply under Dartmouth's Early Decision program, can I apply to other schools under non-binding Early Action?

Depends on the fine print of the Early program at other schools.

If the OTHER school allows you to apply to a binding Early Decision program (Dartmouth uses binding Early Decision) in addition to applying Early at their institution, then yes. The most common example where this is NOT allowed is Single-Choice Early Action, where you can only apply to one school early (though you are not required to enroll at the school if admitted). Remember, under Dartmouth's binding Early Decision program, if you are admitted early you are required to attend (though if you are deferred in early and admitted during the regular process you are no longer bound by the early decision agreement).

Read the fine print of each Early program agreement carefully.