How can a grounded gate MOSFET conduct current?
There will still be a positive gate-to-source voltage if the source is negative with respect to ground. Note that the current source isn't grounded; it's connected to a negative voltage. So there's room for some GS voltage.

EDIT:
Yes, the magnitude of the current through the source can affect other parts of the circuit. It's common to think of \$V{gs}\$ as the input to a MOSFET, and the drain current as the output, but really they're just two variables with a defined relationship to each other. You can put it in a circuit that forces a \$Vgs\$ and allows drain current to adjust to where it wants, or you can put it in a circuit that forces the drain current and allows the gate voltage to adjust. Provided you're operating within the allowable range of parameters of the device, the two variables should always obey the equations. But which is "input" and which is "output" is just a point of view.