Would a helicopter with the blades on the bottom fly the same, if at all?
Yes that is possible, like the Hiller flying platform demonstrated. It had two counter-rotating propellers inside a shroud and the pilot controlled his craft by shifting his body weight, like on a Segway. There is no law of physics that prohibits a helicopter from flying upside down.



The Hiller flying platform was one of several types built in the 1950s, after it was observed that controlling an underslung rotorcraft by weight shift could be learned by untrained personnel in 20 minutes. The DeLackner Aerocycle was another platform built based on this finding. The idea was later abandoned due to practical issues like kicking up rocks into the rotor blades, and an at the time unexplainable interference between the counter-rotating blades. The Aerocycle had a fixed rotor head, the dynamics of which were poorly understood at the time.

For full scale helicopters, having the rotor underneath the helicopter has the same stabilising effect on flight characteristics. Stability of helicopters (and of fixed wing aircraft) is studied from the perspective of aerodynamics: the effects of wind gusts or control inputs.

In the hover, the speed stability ${\delta M}/{\delta{\dot{x}}}$ plays a significant role: if positive, the hover is unstable. A human can learn how to control an unstable platform if the time period of the oscillation is high enough (conventional helicopter pilot), but it is much easier to fly in an aerodynamically stable aircraft (the underslung platforms intended for general infantry personnel). The speed stability effect in the hover equates to a gust of wind blowing from directly forward, does then the change in moment tend to amplify or counteract the effects of the gust. This can be visualised like so:

Image source

The gust blowing on the helicopter tilts the rotor back, which tilts the thrust vector. A teetering conventional rotor has a positive ${\delta M}/{\delta{\dot{x}}}$: it wants to flip backwards ever stronger, and is therefore aerodynamically unstable in the hover. An underslung teetering rotor has a moment change that stabilises: it tilts the fuselage back, and due to aerodynamic coupling the rotor follows back to neutral. A rotor with hinge offset has a smaller stabilising moment, but a stronger fuselage/rotor coupling.

So to answer your original questions: Yes the helicopter with an underslung rotor would fly, and it would be easier to control than a conventional helicopter. It is just a bit unpractical, that's all.