Spoilers here. For Everything. Reader beware...
Watson had an old war, though accounts varied between it being in his shoulder or his hip(ASiS); the narrator has wounds from being cthonically tortured, also in the shoulder(ASiE). The narrator (ASiE) moves in with the detetive at the beginning of the story, having been introduced by a mutual friend working in the chemical laboratories of St. Barts. Similarly, Watson, at the beginning of ASiS, moves is in with Holmes, who was "working in the chemical laboratory up at the hospital," and they are introduced by a mutual friend who also works there. As an incidental, near the beginning of ASiE, Watson takes a hansom, whereas the narrator mentions he prefers hansom cabs near the beginning of ASiE.
The choice of story is itself interesting, as Watson is disturbed by the simian appearance the murder victim of ASiS, and has nightmares about him, much as the narrator of ASiE has nightmares about his torture in Afganistan.
The succession of visitors is in both stories, but whereas ASiS has "a young girl, fashionably dressed," "a slip-shod elderly woman," and "a railway porter in his velveteen uniform", ASiE has "the pale woman with one eye bone-white" and "the portly dandy in his velvet jacket" (parallel with the porter?). In both works, the narrator would, during these visits, retire to his bedroom while the visit occured in the sitting room. And, naturally, both detectives claim to be the only consulting detective in the world. And the use of Rache/Rachel in ASiE is drawn directly from ASiS. ASiS's muderer is "more than six feet high...and smoked a Trichinopoly cigar." By contrast/similarity, ASiE's killer was "a tall man", and smoked a pipe with an unusual blend of shag.
The murderer in ASiS is a cab driver, and has an accomplished actor accomplice. The antagonist in ASiS is an accomplished actor, and evidently has a cab driver ally.
Regarding the royalty in the play:
The Egyptian Black One is Nyarliathotep.
The Black Goat of China is Shub Niggurath.
Not sure of the Czar Unanswerable. Could be Hastur, but given the hints at the end of the story, I'm guessing Azathoth?
Nor He Who Presides over the New World. Probably not enough info to even guess.
The White Lady of the Antartic Fastness is, however, most likely Ithaqua. And probably the Snow Queen.
I'm not sure about the Queen of Albion -- she may be specifically unspecified, as well as drawing on aspects of the actual Queen Victoria, interpreted differently
The chapter ads are, of course, ads from monsters of legend -- perhaps in new professions now that the real monsters have taken over. Aside from #1, which is part of the plot.
#2 is Dr. Jekyll selling powders letting you release your own "Mr. Hyde".
#3 is Dr. Victor Von Frankenstein offering Viagra, or perhaps a limited way of bringing you "back from the dead."
#4 is Dracula offering his own abilities as a form of treatment.
#5 is Springheel Jack (offering to repair footwear).
And, of course, the finale makes it clear that the protagonist is the sniper, Sebastian Moran (which fits him styling himself a marksman and crack shot), and the Holmes-like detective is Professor Moriarty. By contrast, the "master criminal" is none other than Holmes himself, as shown by his tobacco addiction, the fact that the "murderer" is Dr. Watson, and his style of writing. Watson does, in fact, know Moran from Afganastan, though I don't know that Moran's torture is canonical (but then, would it have to be?); I'd likely need to reread the story where Moran (and Moriarty) first appear. The "study" is signed S_ M_, Major -- Major Sebastian Moran! (thanks, glamourweaver)
In ASiS, when Holmes is discusing the possibility of sharing, he says "You don't mind the smell of strong tobacco, I hope?", followed by Watson's "I smoke a strong 'Ship's myself." Similarily, in ASIE, when the actor, is settling down to talk, he says "I smoke a strong black shag, but if you have no objection" and the detective says "Why, I smoke a strong shag myself".
12/31/2010 (barely): khab_rmb notes that Moran and Watson share Watson's noted injury (which migrated, depending on the story, from his arm to his leg). This might explain Moran's arm injury, as well as referring to Watson as "the limping doctor".
Having finished both works, I think it's time to put this to bed.