Question: John McClane's wife calls herself "Miss Gennero." Later in the movie her children are being interviewed on television - but how could Hans Gruber possibly know these are her children? And how does he connect her and the surname "McClane"?
Answer: In addition, with admittedly a bit of guessing, McClane himself admitted to Hans (during the moment they met up on the roof and Hans was pretending to be 'Bill Clay') that he 'got invited to the party by mistake', and that comes after Ellis mucked things up with his act of being McClane's old friend. On top of that, McClane also tells Al that he has kids and to find his wife over the radio, which the film previously confirmed the terrorists can also hear due to using the same radio type. So, Hans now knows McClane is married with kids, Ellis lied about being his friend, and someone else asked him to attend the party. Holly is the only party guest he knows has kids, so once the news program shows said kids, it's not a huge leap of logic for him to fill in the last piece, more so with Holly's panicked reaction and glance to the photo she flipped down.
Question: When John is first on the construction floor, he looks across and the shot lingers on a woman in another building. There doesn't seem to be any real point to that shot - am I missing something?
Answer: When Hans and the gang enter the floor and take everyone hostage, that's when John's phone gets cut out while talking to Argyle in the garage. He then escapes up to the unfinished floors and sees the construction foreman's phone and runs over to try that phone and of course the same thing; the phone is dead. He looks out the window and sees the woman talking on her phone in the adjacent building and it confirms to John that the Nakatomi Towers phone lines have been cut since it's not affecting the building next door.
Answer: The film's script offers another explanation: specifically, it reads, "John watches as the woman casually picks up the phone and begins talking to her friend (or perhaps ordering a pizza?) while strolling around the bedroom like she doesn't have a care in the world. He stares in silent shock at how easy she makes the act look, while right across the street he's unable to do likewise."
Answer: He's waiting to see if she looks in his direction, so he can try to signal her. She doesn't, so he moves on.
If you look closely, she is talking on the phone, which confirms to John that the phone lines had been cut.
Answer: He sees her reaction to seeing them on the news; he can tell she knows them, and from this, it's not a big leap to the conclusion that they are her children. Then, the reporter says to her daughter, "Your mommy and daddy...", but Holly has claimed to be single and her husband is not among the party guests, so he further deduces that she must be married to John McClane, or if nothing else that she's avoided mentioning her husband for a reason. A look at the downturned photograph on her desk confirms his suspicion.