What sustains the rotation of earth's core (faster than surface)?

The same thing that makes the surface rotate: the fact that it always rotated. Because angular momentum is (almost) conserved our planet has no other option than to always rotate. (Of course in reality this picture is complicated by interactions with rest of the universe, but those are just small corrections). You can ask then why did it rotate in the first place and this has to do with the origin of the Solar system.

So, the main question is not what makes Earth rotate but what layers of Earth rotate at what speed. If Earth were a solid body then it would rotate with same angular speed everywhere. But the interior of the Earth is liquid, so the picture is quite complicated. Only thing that is obvious is that every part of Earth must rotate (at some speed or other) because of friction: if one layer (say crust) were rotating and other (say mantle) weren't then after a little while it would start to rotate anyway because the atoms of the moving layer would drag the atoms of the static layer with them.

To say anything more than this one would have to consider precise materials and fluid equations in the interior of Earth. Hopefully someone else will come along and complete the picture.

The earth's rotation is slowing down, due primarily to tidal interaction with the moon (transfering angular momentum to the moon). This frictional loss occurs near the surface. On this basis, if the rotational coupling of the inner core to the mantle is imperfect, it would be expected to lag with respect to the slowdown. I think precessional changes (cycle length roughly 23000 years) are probably more important for generating rotational mistmatches. Then it all gets mixed up with/by thermal convection and magnetic fields as well.

According to our current understanding of the formation of planets, they are created from dust which originates in a previous supernova (or other large) explosion. This is called a nebula.

http://en.wikipedia.org/wiki/Nebular_hypothesis

As this nebula contracts into planets, there are generally two possible cases: that the dust that is eventually going to form a planet is spinning, overall, or not. The most general and likely case is that it is spinning in some form. There is no overall reason why it shouldn't, short of some "magical" combination of canceling factors.

What happens when this spinning dust cloud contracts under gravity? It spins faster (think of an ice skater spinning), due to the conservation of angular momentum.

Therefore, you end up with planets with a relatively fast spinning core.

Now, what keeps this spin going? Fundamentally, conservation of angular momentum. Nothing is keeping it going, and it's actually spinning down. It should eventually stop - in million or billions of years. That's how much momentum is stored in it :-)

Why does the crust spin slower than the core? For two reasons: since things spin faster as they contract, you naturally end up with a variation of speed between the outer layers and the inner core; secondly, the core is actually dragging the crust around, but the Earth is not really a solid so the drag cannot be thought of as a rigid motion.