Pots are just variable resistors. They can affect tone when combines with capacitors to form filters. In the case of guitars, tone pots are configured as a low pass filter and will inhibit high frequencies depending on how the pot is rotated. When it's at full, there will be no attenuation from the filter. When it's at 0, there will be a corner frequency that is dependent on the pot and capacitor values. For example, a 500k pot with a 0.047u capacitor (somewhat of a standard for humbuckers) will have a corner frequency of 1/(2*pi/RC) = 1/(2\pi/5000000.047E-6) = 6.8Hz or so, which means anything above that (essentially everything) is attenuated in some way.
Note that the attenuation isn't an immediate drop-off to 0 at that frequency, it's a gradual slope, so that's why you're still hearing stuff. Reducing the capacitor value will raise the corner frequency; a 0.022u cap would have a corner frequency a bit more than twice that of your 0.047u cap.
Different pots will not sound better or worse than each other. All that matters electronically is the pot value. Sound wise, you'll be better off with better quality pots as low quality ones can be "scratchy" when you turn them. Mechanically speaking, you may prefer pots that are easier to turn, making "palming" the pot more convenient if you want to roll your volume down quickly.
Some people claim that different types of caps will have different tonal qualities. A difference in value will undeniably affect the sound since it modifies the filter, but let's just say tonal differences between different cap types is incredibly subjective. Some people swear by it. Then again, some of the same people bought (and are still buying) "reissues" of "historic" bumblebee caps for ridiculous prices and loved them, without realizing that the reissues are just rewrapped Wesco poly caps.
You might want to experiment with different values of pots, 400k is a good suggestion and not often seen. CTS/Alpha/Bourns are the usual high-quality manufacturers. You might want to check what shaft length you'll need before buying them though, best way will be to get the existing ones out and measure them.