Does the refresh rate affect LCD screens?

Yes, but it is worthwhile clarifying.

LCD pixels get turned on by current and turned off when the current stops. On the other hand, CRT pixels get excited when the electron beam hits them and immediately start to fade.

The implications of this are:

• With a too low refresh rate, a CRT appears to flicker as the brain has time to notice that it went dark. This depends on the individual and which part of the eye sees it since our peripheral vision is more sensitive to light.
• An LCD does not appear to flicker because pixels do not go dark between refreshes. They simply transition from one state to the next.
• The refresh rate of both CRTs and LCDs affect how they render motion. For CRTs, higher refresh rates give smoother motion. This is due to the fading property of CRTs. For LCDs, frame interpolation gives smoother motion. However, this requires drawing more frames, which means increasing the refresh rate. This suggests the refresh rate is responsible for smoother motion for LCDs, where in fact this is because of frame interpolation.
• LCDs have a latency for a pixel to change state. This is known as response time. If the response time is too slow, then some pixels do not have enough time to change; this results in ghosting. The refresh rate however, is not a cause for this.
• It is possible to have ghosting on a CRT as well, but only at very high refresh rates, since phosphors fade quite fast.
• LCDs do not have lower limits to their refresh rates. I once tested a display with 3830×2400 resolution and the first version had a 24 Hz rate. No flicker was noticeable. It was then upgraded to 48 Hz, and then, sadly, discontinued. I guess not enough people paid the 27K price tag!

Yes. But since it takes longer for a LCD cell to depolarize than for a CRT phosphor to extinguish, it doesn't affect them very much.

The 59/60 Hz choice is because your monitor supports a TV-compatible timing of 59.94 Hz, and by design Windows lists a 59 Hz and a 60 Hz mode. Doesn't matter which one you select, as the monitor will output 59.94 either way.

See http://support.microsoft.com/kb/2006076 for an official Microsoft explanation on this issue.