Question: Why does Michael want to kill his niece?
Answer: It's tricky to say, as the films have contradictory explanations, and there are different "timelines/universes" in the series. But in the context of this film, Michael is for some reason compelled to kill his family for reasons unknown. Presumably he's just finishing the job he started by killing his sister Judith decades earlier. (They try to give a more concrete explanation in the movie "The Curse of Michael Myers," but it's... flimsy at best. And is contradicted by the following film).
Answer: Because he's a psychopath. And purely and simply evil.
Question: There's talk of Rachel having to cancel her date with Brady. Why couldn't he just go along with her taking Jamie trick or treating?
Answer: Rachel probably doesn't want to ruin his whole Halloween night by making him a third wheel. She likely figures they can just reschedule and he can find other stuff to do since it's a holiday. There's also the chance they had other certain "things" planned that they couldn't do around Jamie.
Question: Did Jamie acting like Michael at the movie's end happen because of the night's events, or when she touched his hand, or what?
Answer: I think it's really up to the viewer to decide. I personally always took it as she simply snapped and briefly lost her mind and became just like her uncle for an instant. But given she and Michael share a psychic connection in the sequel, I've seen other people suggest that perhaps she was under his "influence" in some way.
Question: How were Rachel and Jamie able to make the connection that Michael was Jamie's uncle?
Answer: It's never explained who told Jamie of her murderous uncle, but Jamie is mocked and made fun of by her classmates for being related to the infamous Michael Myers. She could have been told by her adoptive family, classmates, neighbors, etc, but the movie never provides any definitive answers.
Question: Why transfer him at night in bad weather? That and given his past why not have him cuffed to the gurney and have armed guards there regardless of his comatose state?
Answer: The best in-universe answer I could give you to your first question is that Michael just happened to be scheduled to be transferred at night and the weather ended up being crummy. I've been transferred between hospitals at night before. (Albeit, I'm not a homicidal maniac.) But honestly, the real answer is simply... "because movie." It's a horror movie - it's just more dramatic for the scene to be set at night during lousy weather. It wouldn't be nearly as effective a scene if it was during the day in nice weather. A dark, stormy night is sort-of a convention of the genre. As for the second question, he was severely burned in a fire and has been in a comatose state for years and years. Realistically, it was safe to assume he wouldn't wake up, and even if he did, a normal person's muscles would have likely softened into jelly in the meantime. They assumed they'd be safe... but they were wrong.
The question would be why did the characters transfer him at night in bad weather, not why did the film makers set it up like that. The viewer may thought he or she missed the in-film explanation or was looking for someone with expertise in transferring patients to provide an answer. And again, was there any in-film explanation given or persons with experience in transporting patients like Michael (albeit without supernatural powers). Pointing out the caveat of character's actions isn't realistic because it was scripted that way is fine, but pointing out that a movie is a movie isn't a valid answer (or correction).
I did amend my answer slightly before I saw your response. I really don't think my initial answer was that invalid though. That's honestly the truth - it was done that way for dramatic purposes, and any other answer would be pure speculation.
If no in-film explanation is given, speculation is OK as long as it aligns with something that would happen in real life (although I would suggest saying it's speculation). Sometimes people do ask question about why film makers would do something, and an answer like "to make it more dramatic" would be acceptable.
Question: Jamie is Laurie's daughter and also Michael's niece in the movie. How is this possible? Michael killed his sister at the beginning of the first movie and Laurie isn't even related to him.
Answer: There are multiple timelines in the "Halloween" series, and in this particular timeline Laurie is, in fact, related to Michael. In the original "Halloween II," it is established that Laurie is Michael's long-lost younger sister, who was adopted sometime after he killed their older sister Judith. This movie is a sequel to that film, thus Jamie is both Laurie's daughter and Michael's niece. While the most recent trilogy (starting with the 2018 film) has ignored this fact, most of the sequels have Michael and Laurie as long lost siblings.
Question: Unless he went through adoption records, how would Michael know where his niece lived?
Answer: In Halloween: The Curse of Michael Myers, it's revealed that Michael is under the control of a Druid cult using supernatural powers making him indestructible. It's most likely, they gave him the power to sense his own bloodline.
Michael also first encounters his niece in the drugstore, not at her home. After the encounter, he very well could have simply followed Rachel and Jaime home and learned where she lived that way.
Answer: Thank you.
Rob245