Can windmills allow us to consume (and, eventually, over-consume) the wind as a natural resource somewhat in the same manner that we are over-consuming many other natural resources?
Wind is caused by differences in air pressure, where air flows from high pressure areas to low pressure ones. To "consume" all the wind would require air pressure to reach equilibrium, something that won't happen since the effects of solar heating and cooling from bodies of water constantly introduces differences.
The only effect you might observe if you saturate the environment with windmills is that the upwind ones will spoil it for the down-wind ones, just as a sailing ship can spoil the wind for any directly behind it.
No, because as long as the sun shines, there'll be wind. That's what's meant by calling it a renewable resource.
Any wind moving now, will dissipate to low-grade heat within a few hours or days anyway - the winds of last year have all gone into heat death, and have been replaced by new winds, caused by new temperature differences caused by different solar radiation across the earth, and differential rates of heating between land and sea.
So a wind turbine doesn't really change the fate of the energy in any given wind - either the wind will turn to low-grade heat through friction; or it will get converted to electricity, consumed as electricity, and that end electricity use will, one way or another, almost always end up as low-grade heat, typically sooner rather than later.