Indeed, the Reynolds and Mach numbers should be the same, or at least similar to the actual aircraft, depending on what you want to measure. For instance, the maximum lift coefficient is very sensitive w.r.t. Re and if you add a Mach number above 0.3, compressibility needs to be taken into account as well (even though quite good correction models exist for this). On the other hand, if you want to measure for instance only the zero-lift angle of attack, the Re doesn't matter much.
Anyway, where you support the airfoil doesn't change the Reynolds or mach numbers. In fact, you can theoretically support it at any point. The lift and drag will always be the same for every point; what changes is the moment that you measure. It might be convenient to choose the aerodynamic center which typically lies at around 25% of the chord, since there the pitching moment doesn't change with angle of attack. But be aware that this location can still very slightly move with AOA in practice. For more information: https://en.wikipedia.org/wiki/Aerodynamic_center.