Can satellites go directly over the poles?

Yes, satellites can pass directly over the poles, but usually they don't want to do that. The main reason for a polar orbit is to observe the Earth, but for that they usually want a sun synchronous orbit, which goes close to but not directly over the pole.

Yes, technically, but the amount of extra fuel the satellite would have to carry would increase the mass and expense of launching the satellite. The reason that most rockets and satellites are launched in an easterly direction is because the speed of the Earths rotation from west to east increases the velocity of the rocket. A satellite or rocket would need fuel to alter its orbit to a polar orbit.

A satellite can travel over any point of the Earth's surface. However, many orbits simply are unuseful. For example, any orbit including the geographic poles. Take note that a satellite can only remain over a single point in geosynchronous orbit, and these points are limited to the equator.

TL;DR version: sure they can, but there's no benefit in doing so.

I've never heard anything to the contrary.

And I don't see any reason why that would be a problem.

There might be something worth considering if it were to orbit above the magnetic poles, depending on how the craft is shielded from assorted solar activity, but I wouldn't imagine orbiting directly over the geographic poles would pose any sort of problem, for any sort of reason, no matter how real or imaginary it may be.

Yes, satellites can and do pass directly over the geographic poles and also the magnetic poles.

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Yes, and most on this path are launched southward from a base in Alaska.