Centrifugal force of static observer at North Pole

The centrifugal force is $m omega^2 R$ where $omega$ is the angular velocity which is the same for everyone on Earth, $2 pi / 24 hr$. $R$ is the distance from the axis of rotation, so that would be zero on the North Pole. Thus, you will not experience a centrifugal force on the pole. You experience the largest force on the equator.

You are absolutely correct: the easiest way to see if you are in an inertial frame is to see if objects obey Netwonian laws of motion (at least until we start talking high speed things).

For a better intuition on this, consider the following experiment which is on a more human scale. Find a roundabout on a playground. Stand in the middle, and have someone spin it fast. You will easily feel the effects of rotating motion, including centrifugal "forces." Now, sit in the middle on a chair, and while they spin the roundabout one way, you spin your chair in the opposite direction to cancel the rotation out. You will feel no rotating frame effects, just as you do not feel them over a North pole.

If that experiment isn't intuitive enough, we can bring in an outside influence to make the intuition easier. Have someone hold onto a broomstick over the top of the roundabout's bars, and you hold onto it while sitting on the chair. You can use this broomstick to hold onto yourself so that you always face your friend. Then have another friend start spinning the roundabout beneath you. Physics wise, this is identical to you spinning yourself in the second experiment, but the intuition is a bit easier to follow beacuse we can see rather evidently that we are holding still while the merry go round is beneath us. There is clearly no reason to feel rotational effects in this situation (beyond the minimal torque caused by the imperfect bearings in your seat).

So we see that when we are standing on the roudnabout, we experiance rotational effects. If we are decoupled from the roundabout, and remain fixed in an inertial frame, we do not experience them. Likewise, if we are standing on the North Pole, we will experience rotational effects, such as those which cause the surface of hte bucket of water to be curved. But if we are decoupled from the Earth's rotation, for any reason, we are decoupled from the rotation; we feel nothing.

Minor caveat: Practically speaking, the curvature of the water will be so slight that you will not be able to detect it in the presence of the curvature caused by the surface tension of water in the bucket, known as a meniscus. Its not to say the effects aren't there, just that they are much smaller than we could measure. However, this effect does cause the Earth's oceans to bulge. The oceans at the equator are actually 21km further from the center of the earth than the oceans at the poles!