How does escape velocity relate to energy and speed?
Gravitational potential energy is usually measured as a negative value. We do this because an object that is so far away from a gravity well that it practically is unaware of it shouldn't be considered as having any potential energy. So as $r\to\infty$, $PE\to0$. As an object falls into a gravity well, it loses potential energy, so gravitational PE is a negative value. Since energy is conserved, if you can sum the KE (a positive value) and the PE (a negative value) and the result is $0$, that means there is enough KE for the projectile to reach $r\to\infty$, where $PE=0$ just as $KE=0$. At that point, the object never returns. Thus that KE defines the escape speed. Also, $KE=\frac{1}{2}mv^2$ as you wrote, but $PE=-\frac{GMm}{r}$. The equation $PE=mgh$ is only valid near Earth's surface and it represents the change in potential energy from the surface, not the total potential energy (which is why it is positive, not negative).
If $KE+PE=0$ then at $r=\infty$, the speed will be zero. However, in a more practical sense, you are correct. An object at escape speed in a universe with just one gravity source and the object will never reach zero speed, it will simply move slower and slower forever.
At $r=\infty$, the PE goes to zero. The KE then becomes the only contributor to the total energy. so whatever the total energy of the object is, that's its KE. $r=0$ is a complicated case. For a point source of gravity, the PE would become infinite, but (other than a black hole, which isn't covered by Newtonian mechanics), there is no such thing as a point source of gravity. In the usual case, the Force of gravity disappears at $r=0$, but since moving away from that position would still constitute a gain in PE (as you'd experience an opposing force), it can get more complicated to talk about the PE at or near $r=0$. Generally, beneath the surface of an object, the PE becomes dependent on the distribution of the mass of the object. The KE is again dependent on the total energy. But there is nothing very special at $r=0$. In fact, there are no particularly interesting points anywhere. At every point, the total energy is $E=PE+KE$, PE is a negative value that approaches 0 as $r\to\infty$, and KE is whatever is left over such that total energy is conserved.
Potential energy for a point mass (and also for a sphere) is not $PE = mgh $ (this is special case for a uniform field) but rather: $$ PE = - \frac{GMm}{r}$$
where G is the gravitational constant, M and m are both masses and r is the distance between the masses. (in the case of a sphere, the distance is to the centre of the sphere)
Can you see the answer to your question yourself now?