A stupid question for which there's no sensible answer.

Can the elements in a complicated room cancel/balance themselves off so that it ends up being a good room for sound. For example, slanted, high wooden ceilings and wall-to-wall carpeting, and windows and absorbing furniture all seem as though they add weight on either side of bright or muffled, but perhaps they can cancel each other out?

So close to the truth.

Every surface reflects, diffuses, or absorbs, to some degree or other. A book shelf full of books all the same size and lined up is just another wall. The same bookshelf with books of different heights and thicknesses some sticking out others shoved in, is a quadratic diffuser. The worst sound is a bare rectangular room with nothing in it, because the parallel walls bounce all frequencies creating lots of cancellation and reinforcement and uneven response.

In the vast majority of cases what happens is the midrange and treble ranges are handled pretty well just by normal furniture and decor. What few problems remain are pretty easy and obvious reflections that either can or cannot be handled, usually depending on factors having nothing to do with audio. Windows, doors and spouses being high on the list.

The same physics that dictate the above also dictate why almost everyone has the same problems almost without regard to the room. This is because low bass waves are 40, 50 feet and more. At that scale all our rooms are almost the same. This is why almost everyone has the same bass mode problems.

These problems can be solved with a lot of big expensive acoustic treatments. Or they can be solved with a DBA. Your call.

Either way, what we come back to each and every time, what I said in the beginning, there is no one thing more important than any other thing. There’s more than one way to skin this cat. Or beat this horse to death. As the case may be.....or as usual.... is.

I used to think your question made sense but then tons of personal experience proved there really is no sensible way of saying anything is any more important than anything else. You want anecdotes? We got anecdotes!  

Complete systems. Besides my own, also put some together for others. When you're doing it for others you have to do what they want not what you want. Turns out after doing it a few times this works out just fine. I was still thinking it must be important to budget roughly equal amounts to have things be around the same level for a balanced result. Which does indeed work really, really good. But then for kicks one time I tried a budget ($1200 total) system with about $2k of interconnect and power cord. The improvement was just as big as when those same things were in my ten times as expensive system.

This holds for everything else as well. Is the power cord going to one component any more or less important than another? Not that I can tell. Is the last 3 feet any more or less important than the first three feet coming out of the panel? Nope. What about the three feet inside the speaker? Not even.  

The problem, I eventually realized, is we are asking the wrong question. Who cares which one has the most influence? Which is just another way of asking Which one is most important? Who cares? What if we know for absolute certain it's the speaker. So what? Does that mean we can forget everything else and put all our money in the speaker? Nope.  

What we really want to know is, Of all the things that matter (which is all of them!) what is the one I can do right now, for the least amount of money, and for the most amount of sound quality improvement? This question no one can answer- except YOU! 

Because the answer depends so much on exactly what you have, and that includes your listening ability and preferences. If you have a wonderful system in every way but bottom end it might be subs. Then again the same guy with the same system might not want that at all. In that case it might be something completely different.  

Know thy system. Know thyself. Figure it out. Best answer I can give you.