wow, what a nice comprehensive question. This is something that interests me for several reasons, some historical, some practical.
First, is there a difference between "spinning" and "pivoting." Well, I guess this could get into the "weight v. jin" debate, but, for the moment, it is possible to imagine that rotating the body, whether 45, 90, 180, or 360 degrees, will produce a certain amount of "energy." We can ask, then, whether the creators of the form intended that this "energy" be used, or not. I, personally, and humbly, believe that they did --even if only a secondary source. Since the energy of rotation can be extracted or added to any other "energy" produced, it can also be thought of as distinct from them --though not separate from them, and Steve B. gives a good description of how rotation can be used.
You have several specific questions that probably have different answers based on the circumstances and the intention of the practitioner. You noted:
"I checked Fu Zhongwen’s Mastering Yang Style Taijiquan and discovered that this spin is described as being on the ball of the foot, whereas my memory of Yang Zhen Duo and Yang Jun is that they perform it on the heel"
This seems similar to the "weighted front foot pivot" question. If different family styles have different approaches, it probably has something to do with intention. If one expects to practice on a smooth, slippery floor, then one way of pivoting might have advantages. For example, flat-footed might be fine then. OTOH, on uneven ground, it might not be possible. Spinning on the toe might be preferable, but it is also less stable. So, it depends. However, it is possible to distinguish between "empty" and "full" at the sole of the foot, even while the foot is flat on the ground. (It's a question of "mind", yep). So, imho, keep the foot flat and use your intention. This also has something to do with a point on the "Empty step" thread. A spin is, in effect, a type of transition; and it is the transitions where the applications lie. So, in application, a spin will have to fulfill all the necessary tjq requirements: e.g., "sticking," "following," etc., and not just be a movement isolated from what the opponent is doing. The variety of spin/pivots are, as your post points out, quite comprehensive. Well, there's no 270. Anyway.
"In pushing hands or one-movement application drills, I have spontaneously used my knees to deflect kicks or knee strikes in ways that resemble the first two spins. Could this be what is envisioned in these moves? What about the last two spins? I have heard some suggest that they represent leg sweeps, but their execution seems too high for this."
Well, energy can be generated from the spinning of the "great pole", and there are a myriad of ways to use it. The height of the foot, etc., are things that depend more on your personal flexibilty than on specific usage. What was that story about YLC (?) "shouldering" someonein the knee?
"What about refinements in the performance of the spins? Should we use our eyes to spot a place on the horizon to maintain balance like dancers, or scan our surroundings for multiple opponents? I know of balance techniques from other arts, but am reluctant to use them in the spins if this would detract from the T’ai Chi aspect of the training."
Interesting point, but I think spotting would be most useful in multiple spins --where there's the real chance of getting dizzy. If one develops a style of using an extraordinary amount of rotating, then this is an issue. Well, see bagua.
I wouldn't be able to analyze any of this from the perspective of "jin" or from the history of the form. I.e., how many more or less spins were there in Chen style? What was their application? ETC. But, I will leave that.