Through the Harry Potter series, J.K. Rowling gave us one of the richest, most wonderful worlds ever. And, because it is so significant and treasured, I'm going to nitpick the hell out of it.
1. How come they didn't use any muggle inventions to inform their magic inventions?
The series is set between 1991 and 1998, so it makes sense that wizards wouldn't know that much about the internet and other technological advancements like that, but, by 1994, web search engines were starting to work well enough that some wizard should have heard about it and thought, "Hm, maybe we should come up with a magic way of sorting through the information in all these musty old books?" And let's not even get started on the inefficiency of wizards not having any kind of portable communication device. Sirius Black would probably have survived the series if he'd just had a god damn cell phone.
2. Why wouldn't everyone get a magic portrait of themselves and essentially live forever?
According to the magic specifics behind the enchantements that make it so portraits can talk and move and stuff, a magic painter needs to make the portrait, and only the portraits of the headmasters have the wisdom and knowledge of the portrait subject because the portrait is painted early and the headmaster then imbues the portrait version of himself with all that information. Ok. It's a lot more complicated than it seems at first glance, but still. Everyone should get a magic portrait and spend their lives teaching it to be them, so that when, I don't know, they're murdered and leave behind a confused orphan baby, they could still exist in some form to talk to him.
3. Why would J.K. Rowling even include the time-turner?
It's such a clusterfuck of a plot device. There is no person who read The Prison of Azkaban who didn't think, "Huh. If they can go back in time to save Sirius, why couldn't Dumbledore have gone back to save Lily and James Potter?" The internet is full of some gobbledigook "rules" that J.K. Rowling published after-the-fact, but, come on, just avoid time travel plots, lady. You've already got regular magic.
4. So, was Hagrid's dad just a pervert or what?
Rubeus Hagrid is half human and half giant. His father, the human, had sex with his mother, the giant. If all giants are like Grawp--enormous, stupid beasts--that is one fucked up situation. Just sit quietly and think about it for a little while. Yep. Hagrid's dad went into the woods and boned a monster. Cool.
5. If Fred and George had the Marauder's Map during the events of the Chamber of Secrets, why didn't they use it to help figure out who was opening the chamber?
The chamber doesn't appear on the Marauder's Map, but the bathroom that houses the entrance does. After the first student was petrified, wouldn't you think Fred and George would spend some time scouring the map and notice their little sister a) being conspicuously present for all the subsequent attacks, and b) spending a lot of time in a bathroom that most people find creepy because of the annoying ghost?
6. Why would Hogwarts have students ride to the castle on animals they could only see if they'd seen death?
What is the point of singling out all of the children who have seen dead people and making them feel weird and confused? It's such a horribly morbid way to start off the year.