Wow. What an amazing, rich story. And as good as your discussion was, I feel like you just scratched the surface. I will say that — with all respects to really amazing works like "Island of Dr. Death and Other Stories", "Toy Theater", "How the Whip Came Back" — this is the best of the stories you've covered yet, and the one that strikes me as the most characteristically Wolfean, somehow. As I've noted before, I feel like Wolfe does some of his very best work in his 70s novellas, and this is the first of those. (And The Fifth Head of Cerberus, of course, is three more; and The Book of the New Sun began as a novella intended for Orbit, too.)
Incidentally, minor correction: Orbit wasn't a magazine; it was a series of original anthologies, which were their own thing (some people in the 70s talked about them replacing SF mags), and they came out once or twice a year, as hardbacks. (IMS, later volumes were never even reprinted as paperbacks, part of why the series failed.) It had a few magazine-like features, but wasn't one.
Ok. Let's start with Wad. I can't buy, at all, Brandon's theory that he is on Earth, and doing some equivalent of skyping in. It just raises too many questions about why they don't otherwise contact Earth, or even consider doing so, throughout the story. This is, after all, a first contact situation. If their communications are good enough that they could have the sort of continuous streaming to allow for real-time interaction, wouldn't they, y'know, check in briefly with whatever their equivalent of Starfleet command is? (Also: it's unlikely that Wad deals with multiple captains, not only because he so closely mirrors Daw (not only looking alike, but the name), but also because Daw himself remembers the captain he trained with. There's no reason to assume the training system changed.)
Because of this — and because of the arguments you both made against the clone theory — I, like Glenn, take Daw's description of what's going at face value: Wad is a training program, run by Gladiator, gaining data to be used in training back on Earth. This presses the question: why is he (it?) in this story?
I think the answer is because of some of the themes of this story that you touched on, but which have further depths (this being Wolfe!): namely, simulation. What does it mean to simulate something? At what point is a simulation itself a person? (This is a question Wolfe will return to frequently, with the simulation of Thecla in BotNS (shan't say where, too spoilery), not to mention various things in BotLS, and the basic ontology of BotSS, etc.) Here, the questions of simulations are entwined with the question of communication, and with the question of data modeling (note how much time is spent on data models, such as the different ones Daw and Wad use to model the alien ship). On some level, of course, modeling, communication and simulation are facets of the same complex gem; or perhaps I should say that modeling is necessary for both communication and simulation, and that sufficiently advanced communication may be indistinguishable from simulation.
Some other ways this comes up: the numbers section, that you both were so eager to move past, is all about this: how do you communicate/model (it's some of both) numbers: what is going to be the same in this between aliens and humans. The fact that they find the equivalent to a decimal point and our digits is significant, suggesting that communication and modeling are going to be possible because aliens, and we, are going to be ultimately solving the same sorts of (linguistic, engineering) problems — note the comparisons made in this regard when Daw talks about how they too have handles on their doors, wires, etc. Note that while the simulation of Mr Youngmeadow says they don't know all the words (what the meaning of is is, we might say), he/it also does manage to communicate pretty clearly. Wolfe's story is, I think, both talking about how unbelievably difficult alien communication will be (in contrast to the breezing past this in Star Trek (not all of TNG, of course, but that hadn't come out yet)), but also one that holds out the notion that it is possible. And of course this is the title of the story: alien stones, i.e. the way aliens model mathematics (and the odd, marvelous little mythical notion that the physical stones themselves, wherever they are back in the dust of Earth, somehow matter).
And not just aliens. The difficulties of communications within species — humans' different ways of modeling/communicating come up with Daw and Mrs. Youngmeadow, for instance.
We haven't yet pointed out that the very first words of the story are spoken by a computer — something which is capable of running the ship itself (but not, interestingly, adapting/innovating — maybe why they miss it on the ship?) — and with obvious similarities with the aliens.
I also took — perhaps inaccurately — the quote about marriage at the end to be on this same topic: the questioners were misunderstanding (mis-modeling) heaven, assuming their words and concepts (marriage) apply when they would not (just as Daw cautions words like "robot" might not apply); in other words, heaven is an alien world, which needs to be "translated" (communicated/modeled/simulated) on its own terms, in its own way. Although Jesus also suggests that it will be able to be, and not just by dying and getting there: with sufficient knowledge of scripture and Power of God, they should know this (although maybe the latter is unknowable).
All these rambling thoughts add up, I think, to the idea that this is a story about communication, and about how data is modeled, and what simulation is, and how these are all aspects of the same question, which unites communicating with aliens, and how they understand math, and what Wad is, and how they use Youngmeadow to simulate communication, etc. I don't pretend to know how it all adds up. (My own modeling is inadequate here, obviously.) But I think that's where the story points.