You just asked almost all the hard questions in room acoustics, without even pausing for breath
1) A perfectly square room is bad, but a perfect cube is worse.
Yes. Having one dimension be a multiple of another is almost as bad. This is only a problem in the lower frequencies, where the room itself acts as a resonator and emphasizes certain frequencies over others. The result is a boomy sound at the emphasized frequencies and "weak bass" at other frequencies. Bass traps (see 3) can lessen this effect, as can (sometimes) the use of an equalizer to reduce the peaks.
2) It is acoustically beneficial to have some sort of sound absorbing material at the first reflection points.
Yes. Here you are trying to ensure that the sound from the speakers reaches your ears without being immediately confused by "close" reflections. If there is enough delay between a sound and the echo your brain seems to figure it out when determining where a sound is coming from, but reflections which are very close in time to the original sound muck up your perception of "imaging", "soundscape", that kind of stuff. You are mostly worried about mid and high frequencies, so a couple of inches of foam or rigid fiberglass is just fine. Note that carpet tiles will probably absorb the high frequencies but are probably too thin to absorb much in the midrange frequencies.
3) It is acoustically beneficial to have some sort of sound absorbing material at the corners of the room. (or any corner for that matter)
Here we are talking about bass traps -- absorbing the lower frequencies. You need big thick absorbing material to absorb bass, so bass treatments tends to be big and, accordingly, expensive. Bass traps can go anywhere but are more effective in corners -- people put them in corners so they don't need as many of the big ugly things
4) It is also acoustically beneficial to sound reflective materials throughout the room.
Yes. The trick is to get the right mix of reflection, diffusion (bouncing in different directions) and absorbtion. When it comes time to do my room "right" I'm gonna call Rives and have them tell me what to do, since that stuff is way above my pay grade. In the meantime, just try to have a nice mix of surfaces and "stuff" in the room and don't worry too much.
Again, note that if you ONLY use carpet tiles for sound absorbing you will get a slightly wierd sounding room, where the high frequencies are absorbed but the midrange frequencies are still bouncing around the room like one of those old "superballs".
5) There are mind boggling logorhythmms and calculations that make this crystal clear to some, and more confusing to others.
Partially right. To the best of my knowledge this is not crystal clear to anyone. Even the best acoustical consultants only claim that the most basic room shapes and materials are crystal clear. They have been doing this for 25 years and still get stumped every so often.
If you read enough and do some experimenting you can go from stumbling around in the dark to stumbling around with very dim light... but that dim light makes a huge difference
