Question: Why did Jasper need to pay Buck to let him rape The Bride? Why couldn't he just walk in and do it for free?
Answer: Buck is wearing scrubs in the scene, so he probably works at the hospital. (A Kill Bill wiki says that he is a nurse.) Jasper is paying Buck for access and to not be disturbed while he commits the rape.
Question: Why did Matsumoto have O-ren's parents killed?
Answer: The film does not explain, but O-ren's parents must have been involved in some sort of shady business dealings to have incurred the wrath of Matsumoto. When he is killed, O-ren's father is wearing the uniform of a United States Army Sergeant Major (pay grade E-9), which is the top rank of an enlisted man. He must have been working with Matsumoto in some sort of illegal smuggling ring that involved international connections through the U.S. Army that went bad, or maybe O-ren's father wanted more money and he was in the way, etc.
Question: In the scene at the House of Blue Leaves, (right after the Bride kills GoGo), O-Ren is holding Gogo's small sword. She begins to unsheathe it before stabbing it into the railing. What was the point of it? Was she upset because Gogo died, or was she contemplating suicide in lieu of facing the Bride?
Answer: To be honest, it's a mix of both. O-ren and B were close in the hit squad. B knew O-ren's backstory, suggesting a shared past. They had common interests, spoke Japanese, and used samurai swords. O-ren confided in B and may have been her squad member. O-ren shared her shame with GoGo, indicating a deep bond. Bill supported O-ren's Tokyo takeover, indicating their closeness. Their shared dialogue suggests an inside joke. O-ren contemplates suicide for betraying her friend and killing her child. She feels remorse for failing to defeat Beatrix and cares for GoGo, who died protecting her. When the 88's arrive, O-ren resolves to fight for honor and avenge GoGo. She knows Beatrix will defeat the 88's, so she prepares for their upcoming battle.
Question: If Bill is behind the death O-Ren's parents, did she know? If so why didn't she go after Bill?
Answer: There is nothing in the film that states or even particularly indicates that Bill is somehow behind the deaths of O-Ren's parents. The only explanation we get is that their death was ordered by Yakuza boss Matsumoto, who brought in the thugs that killed her father. There is a semi-popular fan-theory that the man in white (Pretty Riki) is actually a young Bill, but to my knowledge, this was never confirmed by Quentin Tarantino. (In fact, according to the Kill Bill wiki, Tarantino actually denied they were the same person, but I can't find the source for that.) So there's literally no reason for her to go after Bill. As far as she (and the audience) knows, he was uninvolved in their deaths.
Answer: She didn't, she tried right away. Then Bill called her at the last minute and stopped her, out of respect for the Bride. It's only after this that we cut to four years later, when she wakes up.