As the main page mentions, the live performance IBC is broadcasting is officially "Scrooge", not A Christmas Carol. There are at least two possible in-universe reasons for this:
The Show Within a Show is a loose enough adaptation that they decided to not officially refer to it as Dickens's work: From what we see of the production, it does stray far enough from the source material to involve the Solid Gold Dancers, Tiny Tim doing gymnastics, gratuitous cleavage in costumes, and a dormouse with antlers.
Viewers Are Morons: The film has a pretty cynical view of network television, so it's possible someone at IBC decided that the name of the main character of A Christmas Carol is more recognizable to the general public than the name of the work he's from.
Or, conversely, it could just be because Scrooge is not an uncommon title for adaptations, including such well known examples as the 1951 Alastair Sim version and the musical version from 1970. And using the alternate title for the in-story production helps tie it in better with the film's title of Scrooged.
One might wonder why Wendy of all people is crying at Frank's funeral when, by all accounts, Frank never really talked to her except maybe occasionally through his brother. But then when his brother walks into the scene, it becomes clear how thoroughly heartbroken he is at his brother's passing, and you realize she's not crying for Frank, she's crying for James.
The speech about Frank not being able to work late without Grace working late seems funny and hammy at first. However, when the Ghost of Christmas Yet-To-Come shows what became of Grace's son Calvin; we see that she is kicked out of the institution her son was committed to as soon as she arrived. The reason being that visiting hours is over. This implies that even after Frank's death, his successor forces her to work late so she can't even visit her son in the hospital.
Or, given the points the Ghost takes Frank to might not be at a consistent time in the future, Frank himself could be working Grace even harder than he is in the present. That would certainly add a bit more guilt to it and extra cause for Frank to say he can fix things, particularly as he already feels bad from learning about Grace's life in the present.
Claire's number was saved in Frank's office phone (the phone displays her name, which it only would with numbers saved). He's not talked to her in fifteen years, and it is entirely improbable he's had the same phone since 1973. Ergo, Frank is not entirely heartless even before the ghosts come knocking. He's kept the number around, for whatever reasons a Bill Murray may have. As for why that number hasn't changed in fifteen years... that's another question altogether.
Claire's a charity worker/organiser, meaning her income can't be that fantastic; It's possible that she still lives at the same apartment we see her and Frank in when he's taken back to the past. As for Frank not being entirely heartless - that's a given, otherwise there would be no going back for him to begin with; he's just not terribly considerate of others, to say the least.
When observing the games his brother and his friends are playing alongside the Ghost of Christmas Present, Frank is surprised to find that his brother doesn't know the name of the ship from Gilligan's Island, and later calls him out on this during the finale. But of course Frank would know - he's the kid in the family the Ghost of Christmas Past calls out for doing nothing but watching TV across his youth, meaning Gilligan's Island is exactly the kind of show a teenage Frank would have watched.
"Take it easy on the Bacardi?" Wouldn't someone as loaded as Frank keep some higher-end booze around? Probably, but, much like Scrooge himself, Frank's a greedy miser, so of course all he'd have set out is the cheaper stuff.
The reason why Frank is dead in the bad future is because Loudermilk killed him in the latter's rampage. After experiencing the bad future and rehiring Loudermilk, he prevented his own death.
Unlikely. James and Wendy are visibly older than in the present during the cremation scene. Considering what a Bad Boss he was though, it's not impossible to connect that to his death via another attempt.
If Loudermilk's wife left him on Christmas Eve, just because he was fired from the job that he'd only just gotten, either she wasn't much of a wife or he wasn't much of a husband. Considering his response to this event is to get raging drunk then break into the office of his former boss and try to take him down with a shotgun while singing "Santa Claus Is Coming To Town", then holds the office hostage on command as soon as he gets his job back. Even though he was originally fired for standing up to his boss, it's not implausible to believe it's the latter.
