As the earth's core cools, will the size be affected?
Over the 4.6 billion years of the Earth's evolution, so far, the inner/outer core interface has "cooled" to a mere 6000 deg C. Of course, other things being equal, the core will contract as it cools, but the rate of cooling will be massively delayed by the slow radioactive decay of uranium 235, U 238 and Thorium. Delayed, in fact by many more billion years. The mantle will cool with it, and being plastic, will deform to take up the space of the thermal contraction. However, by the time the radioactive heating decays to much lesser values, the sun will have blown up into the red giant phase. Even if the Earth is not actually engulfed by the expanded sun, it will be so heated that the oceans, and more volatile fractions of the crust will have boiled off into space, to become  part of the enhanced solar wind. 

If we assume that the Earth's core will eventually cool by 4000 deg C, then Earth will eventually have a reduction in radius of about 10 km (my very rough back of an envelope calculation) from thermal contraction alone. How much of the surface will have boiled off into space by then is anybody's guess.