is a subwoofer for music a good idea??

Get the best used sub you can for that price. Try it for a couple of months. If you don't like the level of quality it provides sell it and keep saving your money for a better one. But the answer to the question of whether or not to get one is a slam dunk... yes!

It's not just about reproducing the specific low fequency instrumental sounds. It's also about energizing the room. A good sub, especially if you are using monitor speakers, will allow your music to float in the air, it will make the whole experience more musical.

Look for one you can adjust for frequency response, phase, and "slam." Then plan to spend hours adjusting the location and other parameters to optimize the sound for your unique room and personal listening tastes. When you get it right, you'll know it was worth the trouble.

I have been particularly pleased with my REL subs.

Subs and mains should not overlap. They should be set up to merge at a point where they are both 3 to 6 db down. Combining the output of the two sums them together and produces flat response. Overlap creates a big hump and sounds muddy and bloated. Keep subs crossed very low and just loud enough to contribute impact and weight as needed. Adjust phase according to placement.

Depends on the music. Some music (Classical chamber music)leaves the SW completely quiet, while some (organ music) keeps it busy. Pop/Rock benefits from a SW. Most Clasical music does not require a SW, although it is occasionally helpful.

And what frequency range are we talking about? There really isn't much below 30 Hz, so a SW that is good only from 30 Hz up, and perhaps is therefore capable of 200-300 Hz ("fast")may be a good idea. Getting the 100 Hz-and-down out of your main speakers and their amplifier is the greatest benefit.

Although some main speakers may reproduce 40 Hz, this does not always mean that they have a "full" sound in the sub 100Hz range. When you get a SW you may be discouraged that so much of the time that big expensive object it just sitting there like a rock, doing nothing. But it's like the fire department: you don't need it often, but it's nice to have on hand when needed.

Eldartford,
I think that Beethoven's 5th would benefit from sub.
Also having a vinyl playback it's also a large benefit to have a sub especially dealing with warped records.