In the order of your questions:
1. Yes -- a subwoofer can add a lot to the reproduction of music, PROVIDED the sub is of good quality and is compatible with your main speakers. To get a fairly seamless blend between the main speakers and the sub, there should be about a one octave overlap between them, with the crossover point for the sub around 80 Hz.
2. Cost doesn't necessarily equate well with how low the sub will go, although there is certainly some correlation. There are some decent powered subs that sell used for $400-500. What you need to focus on is finding a sub that is reasonably well damped (probably a sealed enclosure), and has 2-3 smaller drivers (which have the ability to move a fair amount of air, but still are quick and tight). The placement of the sub in your listening room will also have a MAJOR impact on the sub's performance.
3. There are certainly frequencies below 40 Hz in music, but not as much as some people may think. Commercial rock and pop recordings, for example, do not have a lot of really low frequency, whereas jazz and classical recordings with acoustic bass and piano have frequencies that may go as low as 28-30 Hz. Pipe organ recordings have the lowest frequencies, which may go as deep as 16-18 Hz.
4. This question is the toughest to answer -- the best I can say is "maybe", but much of this will depend on the quality of the sub and its placement in your listening area. With a single sub, it should be placed between the two main speakers, and if you have the budget and the space then you should seriously consider a stereo pair of subwoofers. Good subs, properly placed, will have very little affect on imaging, and may also have the unexpected benefit of improving the mid-range transients and clarity of your main speakers (since they are not reproducing the lowest frequencies).
You have a few choices to make: do you intend to get a powered subwoofer, or a passive unit and provide your own amplification? If your budget allows, a powered sub makes more sense, since the internal amp is designed specifically for that sub. There are some good passive subs, however, such as those made by Hsu.
One of the most musically satisfying subs on the market is the Vandersteen 2Wq, and is works well with many speakers, provided the main speakers have decent frequency response down to about 40 Hz (to provide the needed one-octave overlap with the Vandy sub). There are a number of articles, some with ratings of subs, available in the A-gon archives, and you might find it helpful to do a little reading through the old posts here.
