I was working with an interface with a MOSFET and I ran into a case where high impedence might be an input to the MOSFET's gate, essentially leaving the gate open. Would the MOSFET exhibit the same electrical & physical behavior as if the gate was connected to GND?
Not even close. The complete opposite, in fact.
A high-impedance floating N-channel gate can pick up charge from a variety of sources, (via the parasitic gate-to-drain capacitance, via handling - touching the gate with your finger, etc.) and spontaneously turn itself on. Any gate that could have a high-impedance control (a digital I/O from a micro, for instance) really needs a gate-to-source resistor to bleed off these spurious charges and keep the device state defined (=off) when the control is high-Z.
No, it will not behave the same. It depends on the local environment and local leakage currents, stray electric fields etc. In short it's not very predictable at what state the gate will be and then subsequently what state the channel will be underneath. The bigger MOSFET's having larger gate capacitances will respond more slowly.
Put in a high value pulldown/up resistor to deal with this High "Z" state.