With regard to your second question, see:
http://macro.lsu.edu/howto/solvents/Dipole Moment.htm Compare isopropyl alcohol to acetone.
With regard to your first question, it depends on how you define polarity. If you define it as dipole moment, your argument isn't really going to work. But as chemists, we usually define polarity in terms of observed phenomena. For instance, boiling point. Or solubility. And the problem with that is that something with a lower dipole moment but with the ability to hydrogen bond can still have a higher boiling point than something else that has a comparatively higher dipole moment but without the ability to hydrogen bond. If you define polarity as retention time on a silica column, well that's a problem too. Because hydrogen bonding largely determines that. So if you want to define polarity as a function of boiling point or retention time on a TLC, that's fine. But a dipole moment argument would not work.
Just consider this - let's go back to first principles. So you have cyclohexanone vs. cyclohexanol. Now, the oxygen on the cyclohexanone has two bonds to it, both to carbon. They are oriented in the same direction, 0 degrees with respect to each other. So you get the full force of the carbon pulling on the oxygen. In cyclohexanol, you have two bonds to oxygen, one with carbon and the other with hydrogen. These are oriented at 109 degrees relative to each other. So the dipole moment is diluted, so to speak, and only one component of the dipole vectors will play a role.