The opening baby-carriage scene seems incongruous, considering the Big Bad of the film wants Oscar alive. But if you think of it as something the little bit of "mood slime" that got on its wheels did on its own, it makes sense: it only starts to move when Dana's building super agrees to help Dana with her grocery bags. As it'd already picked up some negative energy from the bickering people she'd walked past, it was repelled by the man's helpfulness and moved the carriage away. Then Dana saw it'd moved, and her own Mama Bear feelings for her child both repelled it (her love) and charged it up further (her fear), keeping it in motion until it got far enough ahead to be out of range. Luckily for Oscar, that was before it could roll in front of the oncoming bus.
Wexler, despite his reputation and temperament, seems to go way above and beyond in his condemnation of the Ghostbusters when sentencing them (which any competent lawyer would use on appeal to have the conviction thrown out). Seems he should know better, until you remember that he is in proximity of the negatively-charged slime...
While in their "construction worker" uniforms, Peter, Ray, and Egon don't seem to be recognized by the passing police officer. Indeed, the former two are even bold enough to call each other by name...but Peter calls Egon "Ziggy", for some reason. Rather than an odd Non Sequiter, there might be some logic here: there are probably lots of Pete's and Ray's in New York, but the name "Egon" is slightly harder to come by. The officer might not recognize the guys by face, but if a distinct name pops up, he might be able to put the pieces together faster and move to arrest the guys earlier than he did.
Ray and Winston carry the Slime Blowers because they are the most positive Ghostbusters, as opposed to Egon, who is borderline emotionless (being seemingly immune to the same slime that made Ray and Winston want to kill each other), and Peter, who is...well, Peter Venkman.
Upon study, the Titanic arriving shouldn't be possible since the ship split in half while sinking. That said, the supernatural energies unleashed likely repaired the boat, save for a large hole in the side which, depending on how the boat was moved, was either left for water to drain out or is simply wasn't fixed before docking.
Another interpretation is that the ship "died" before it snapped in half. It arrived in New York in the state it died in. It also got the wrong pier because the harbor had changed significantly since the time of its journey, and the pilot didn't come out to meet it.
The Titanic being in the harbor, as well as the Statue of Liberty walking through New York proper, explain why the Ghostbusters aren't sued again. New York somehow rationalized Gozer's attack as a giant con, but they can't explain how the boat and the statue could be moved scientifically.
When the Ghostbusters charge up their proton packs again, they each start saying the first 3 notes of "Do-Re-Mi". Venkman starts with "Do...", Ray continues with "Re" (smiling at the obvious pun with his first name), and Egon says "Egon" rather than "Mi", because... well, in that case, "Me" was Egon.
Venkman's insult towards Vigo that "only a Carpathian would come back to life now and choose New York", insinuating that this was an extremely poor choice on Vigo's part. Well, it was. Vigo's plan hinges on using the psychokinetic energy of negative emotions to boost his power to a peak on New Year's Eve in New York, a time and place that has been a stereotype for party central for decades. Sure enough, midnight hits and the usual New York New Year's Eve party atmosphere severely weakens Vigo right when he was counting on a power boost.
Vigo's bloody history is alluded to by himself and several other characters in the movie. While it initially seems like the standard reputation someone with a haunted painting might have, meant to set Vigo up as a formidable villain, it also explains a little bit of his implied relationship to the negatively-charged pink slime: Someone with a violent and oppressive regime would be creating prime conditions for the slime to flourish in.
