2, 6 and 8 if you feel like it ;w;
Thank you so much for the ask!! I apologize for not answering sooner. I was asleep when I got the ask and have only gotten to my computer now.
2.) What Crusader are they closest to? Alternatively, do they have any ‘beef’ or rivalry with any of their fellow Crusaders? If so, how and when does it get resolved?
The Crusaders that Barbara ends up being closest with are Jotaro -- whom she's known since they were kids because of their parents being close -- and Kakyoin -- primarily due to being close in age. She and Kakyoin become partners-in-crime on some level. Even if their first true interaction with each other is Kakyoin pretending to not understand English to mess with her a little, and Barbara catching him off guard by being able to talk with him in Japanese.
The only "beef" she truly has is with Jotaro. And that comes from not seeing or talking to each other in over two years due to outside issues. Barbara had family issues with her sister, and this stems from a headcanon, but Jotaro's delinquent phase started when he reached high school. They don't meet face-to-face again until Jotaro gets arrested. How much each has changed throws them around so much that they can't just jump back into how things used to be. Things smooth out slowly over the course of SDC, though.
6.) How did your OC join the Crusaders? Did they join during a specific arc or were they there from the beginning? What drives them to want to aid in defeating DIO?
Barbara was there from the beginning. Due to her ability to sense Stands and Stand Users, she was brought to Japan when Jotaro was arrested to make sure that it was his own Stand that was the "evil spirit" possessing him and not an Agent of Dio.
Barbara's drive to want to defeat DIO, though, has to due with something that happens shortly before SDC begins. Her mother was killed by DIO's agents for being a Hamon User, while her father -- a Stand User -- went missing and was presumed dead too. Barbara's mother was close to Holly to the point that they were like sisters. Barbara doesn't want anyone else to die. She doesn't want to lose yet another mother in her life, and she doesn't want Jotaro to go through the same thing that she is.
8.) Which canon death hit your OC the hardest? Does your OC survive the journey themselves?
I don't know if this is just a me thing or what, but if I can find ways for characters to survive, I'll take it. So in the timeline that Barbara exists in, the only canon death I have yet to avoid is Iggy's. She was close with that little dog, so the fact that he dies protecting the rest involved in the fight hits pretty hard. If the canon deaths did remain in her universe, it's actually Avdol's death that would hit her the hardest. When everyone gets separated, she's with Avdol, Iggy and Polnareff for the Vanilla Ice fight. It would tear her apart that Pentacle Queen wasn't able to catch Vanilla Ice in time to save Avdol.
She does survive to the end of Part 3. When the dust settles in Cairo, she comes out of it with a pair of scars across her neck and a shit ton of blood loss from DIO taking a sip.