plug a lan cable into a wireless extender? uh nope, never heard of it being done, but I guess I never looked them up. But yes, they really do work.
But what works just as good if not better is buying an "N" capable router and adapter (and either canceling the ability for "g" capable or lower to connect, or not connecting any.
wireless extenders are made to carry the WIRELESS signal further in a building/outside, they do not offer any other benefit