Since you had a battery powered thermostat, I will assume the white wire was used for the heating circuit. But you need to always verify that the color of any given colored wire is connected to the same designated terminal (R, W, G etc) at both the thermostat and control board at the furnace. In your case, to have all the same functions (heat, fan, cold) you will need a 5th wire to bring the common terminal at the furnace control board to the common (C) terminal at the thermostat for its power.
As for measuring voltages between the transformer hot (R) side and other connections, depending on what safety switches are present and their normal open/closed state, you may or may not see a voltage differential. It is true that all the other terminals (W, G, Y and C) eventually connect to ground, known also as the common and being the other side of the transformer secondary. However they connect through a series of devices including semiconductors and coils and switches. All these devices have impedance and therefore will cause a voltage drop, assuming some safety switch isn't normally open in which case you'll see no voltage drop... it will look like an open circuit, since it is until the switch closes.