How does intellectual property work?

Copyright and trademark are totally separate--don't confuse them. Anyone who comes up with a new graphic (picture or image) or verbal (words, such as a written quote or saying) expression, work, content, etc. owns the copyright to it even if they do not register copyright and has the right to stop other people from reproducing, using, etc. the copyrighted material. (Note: when we say that the person who came up with the expression owns the rights, the right could also or instead be owned by their employer or publisher, depending on the arrangements among them, but the point is, the act of creating a new graphic or verbal expression or work gives *someone* the rights to it.) So regardless of whether something is trademarked, you can't use someone else's original verbal material without permission, which generally means without paying them a licensing fee.
The de minimis use exception is very tricky. To oversimplify: if you what intend to use is a very small ("de minimis") part of a longer work, it can often be used legally. But first, it has to be a small piece of something longer; and second, if the small piece you want to use is the most famous or identifiable part of a longer work, then it is not necessarly "de minimis" or unimportant. Recognizable or famous quotes will often not be de minimis because the quote is either not taken from a longer work--it is it's own thing--and/or because it is the important part of something.
Copyright does not last forever: to again oversimplify, assume it lasts around 75 years. That's why you had the book and movie "Pride and Predudice and Zombies": the original "Pride and Prejudice" is so old, anyone can use, modify, adapt, etc. Jane Austin's text.
As to the websites to which you refer: some of the quotes may be de minimis; some may be old enough to be out of copyright; and some may be infringing copyright, but the owners of the copyright either haven't noticed or don't care enough to do anything. Take YouTube: it has music videos, songs, song compliations, albums, TV episodes, and movies on it. Many of them are *clearly* under copyright--but if the copyright owner doesn't notice it or doesn't choose to tell YouTube to remove it, it stays up. The YouTube clip is violating copyright, but the person who posted it gets away with it until someone with the rights complaints and takes action.
To what what you want to do correctly or legally, take your proposed quotes and their sources to a copyright attorney and review with him or her which you can use, under what circumstances.