Big Lipped Alligator Moment: While Batman and Robin beating up some bad guys during the 'Dynamic Duet' sequence. Robin needs to run into the corner and cry hysterically for a few moments, before continuing to sing with Batman. This is never explained or mentioned again.
From the same song, all of the villians (who previously simply seemed to double as background dancers) suddenly applauding Batman and Robin and giving them a big hug, only to go right back to trying to kill them in the next act.
"The American Way" starts out making sense in the context of the story, as Superman has just been given a mission by the President of the United States, but Batman and the villains have no reason to sing about America whatsoever.
Batman devotes an entire verse of "Dark, Sad, Lonely Knight" to rambling about an ex-girlfriend and his food allergies.
Fridge Brilliance: Alfred says while he shielded Batman from all the bad things in the world, he accidentally kept all the good things away from him too, "Like the birds and the sun..." When Batman and Robin sing about their newly formed friendship, what is one of the things Batman refers to Robin as? His morning bird.
Similarly, Superman is strongly linked to the sun so it could refer to Robin and Superman.
When the reporters are asking Commissioner Gordon about information on the Batman, one of the reporters, played by Brian Holden, is wearing a white button down shirt, a red tie, and thick framed glasses. These are the same glasses Batman uses to try to expose Superman's secret identity later in Act 1. Clark Kent was trying to find out more about Batman.
During 'Robin Sucks', one of the characters uses Stan Lee's catchphrase "Excelsior". That makes sense if you know about Stan's dislike for teen sidekicks.
Fridge Horror: If one isn't acquainted with the original comics, it can be somewhat disturbing to find out that in canon, Robin was essentially Batman's adopted, 12-ish year old son, considering how much Ho Yaythere is in this musical
Fridge Logic: If Superman followed Batman home the first day they met to learn his secret identity, then he must have known that the Bruce Wayne tweeting about how dumb he is was actually Batman.
Hilarious in Hindsight: Several bits of dialogue and plot points in The Dark Knight Rises are oddly similar to this musical.
Ho Yay: It's a parody musical of Batman that has the adoption of Robin as its main story. Does more really need to be said? ...Yes, actually, because much of the rest of the play consists of attractive men running around in tights and underwear.
Batman also has some very... 'close' moments with nearly every male main character in the cast. Viewers have commented they almost expected Batman and Superman to kiss at one least one point
They also rip off each other's underwear during 'To Be a Man'
'The Dynamic Duet' Up to Eleven. In addition to featuring Batman and Robin dancing around in spandex and singing about how much they care about each other, Robin having an interestingreaction to Batman touching his cheek, the song ends with Robin jumping into Batman's arms and kissing him on the forehead
When Alfred gives Batman advice on whether he should save Gotham or Robin he tells him a story from when he was in the war about a man who had to make a similar decision about his wife
Batman refers to Robin as his 'life partner' at one point. Although this could refer to their position as Heterosexual Life-Partners, it's not uncommon for people in Real Life to use it to refer to spouses in a same-sex marriage.
When Evil-Sherlock Holmes is shot in the knees by Batman he laments that he will "Never be able to walk Evil-Watson down the aisle!"