Sorry if the title is misleading or confusing, but I'm glad it got your attention.
What I'm trying to ask is: say you're looking to build a 2.1 system for music playback. When you're considering a sub, do you want the best rated sub or is it possible to consider subs based on their, for lack of a better term, "musical quality" or how they blend with the L/R speakers (if that is even a remotely technical way of describing it)?
It depends on who is doing the rating, and how they rate the subwoofers. You want a subwoofer that has a flat frequency response, is capable of high output at all of its frequencies while having low distortion. This is ideal, regardless of whether it is for a "music" system or a "home theater" system. A perfect subwoofer is perfect for both.
Beware of BS about subwoofers and other audio matters. Usually, BS is close at hand when people start talking about a piece of equipment that is "musical".
If you're listening to music, do you want (or need) a sub that can do 15 Hz?
It depends upon what you listen to, and what you want. If you listen to the types of music I listen to, you need it. Pipe organs can go about that low, and if your subwoofer cannot do that, it will never sound right (or, at such low frequencies,
feel right) when trying to play music that uses the lowest notes of a pipe organ.
But, for most pop music, 15 Hz is probably irrelevant. Most people probably don't need a subwoofer that goes down to 15 Hz.
Is the playback quality of 2.1 music going to be a reflection of how low the sub can go or other things like crossover settings and room acoustics?
The quality of the rest of the system is irrelevant to the performance of the subwoofer, unless, of course, the source does not have a flat frequency response at the subwoofer frequencies, or has massive distortion or noise at those frequencies. In other words, for all decent equipment, the other equipment is irrelevant for the subwoofer performance.
The main speakers should be able to go low enough that there is no gap in frequencies covered by them and the subwoofer. I suggest using main speakers that can at least go solidly down to 80 Hz, which means the - 3 dB point should be somewhat lower, let us say 60 Hz or lower. I think a crossover setting of 80 Hz or lower is best, because going much higher will result in you being able to locate the subwoofer from the sound alone. But, of course, if one's main speakers were only flat to 100 Hz, then a higher setting would be needed for a flat frequency response.
Room acoustics and room placement will matter. Typically, front corner placement is best, but it depends upon the particular subwoofer and the particular room. This, by the way, is one of the nice things about having a separate subwoofer: You can put it where the bass is best, and the main speakers where other frequencies are best. Of course, this is one more thing to figure out where it goes, and then, of course, one must select an appropriate crossover frequency, and set the level of the subwoofer to match the level of the main speakers. There is more that one can mess up with a separate subwoofer.
From rereading your question, I may have misunderstood you the first time. The playback quality of the system will depend on how low the subwoofer can go as well as other things like crossover settings and room acoustics.
Maybe this all sounds confusing from a videophile/audionewb like myself, but can subs suffer from things like coloration or poor frequency response as much as loudspeakers can?
They can suffer from poor frequency response, and many of the cheaper ones do (and so do some of the more expensive ones; price is not the same as quality). They can also distort massively, though studies have shown that at really low frequencies it takes incredible amounts of distortion for you to hear it. This is because human hearing is most sensitive to midrange frequencies, and at the extreme limits of one's hearing, one cannot hear very well. Still, you want to keep the distortion as low as reasonably possible.
Is carefully picking a sub for 2.1 music less important than carefully picking your L/R loudspeakers (I suppose that is concise enough for a coherent question

)?
Yes, less care is needed for selecting a subwoofer than for the main speakers. This is because the main speakers cover a much wider frequency range, and they also cover the frequencies that you can hear the best. So imperfections in the main speakers will be more audible than imperfections in a subwoofer. This is also illustrated by the fact that you only need one subwoofer, as you cannot tell, from sound alone, the direction of deep bass. But you have comparatively little trouble telling the direction in which midrange is coming at you. Still, you should take some care in selecting a subwoofer.
Is it more important to devote more of a budget towards loudspeakers versus a sub?
That depends upon what you want, and what part of the frequency spectrum matters most to you. I think a great 2.1 channel system would be had with the following:
http://www.kellsieavdesign.com/products/Leisure2SE.htm
http://www.svsound.com/products-sub-cyl-pcultra_new.cfm
In fact, I would rather have the above together than the following alone (which costs about the same):
http://www.kellsieavdesign.com/products/MG.htm
I am sure that you will be able to find those who would disagree with me on this. And certainly, if one did not set up the 2.1 system properly, it would sound worse than the 2.0 system. But if it were set up properly, I think the 2.1 system would sound better. Certainly, there would be deeper bass.
I understand having a monster sub is great for movies and LFE, but is it a pre-requisite for great musical playback?
Define "great". In my opinion, a great subwoofer is essential for great musical playback, unless one's main speakers are flat down to 15 Hz. But remember, I listen to things with notes that go lower than 20 Hz, so my opinion on this is not going to be the same as many others who do not.