Oh, what a beautiful question! Thank you for asking haha.
So fanfiction is just fiction, to me. For as many original novels as we have out there, fanfiction has always been a massive part of culture. Dante’s Inferno is Bible fanfiction. Cultures retold their myths over and over again. It’s a form of art and storytelling that has been around for basically forever. I write original fic too, but fanfiction I don’t think of as lesser writing, and I think the culture that mocks it is stems from misogynistic attitudes towards fandom.
For me, my favorite part of any story is the characters. Hence, when I write fanfictions I try to focus on them and their development. I often write ships because shipping is fun for me, but I’m working on one right now that is gen and for some characters I only want to read gen. I guess that’s because it comes down to the character.
I like AUs because I want to explore characters in new settings, because I think while some parts of them would clearly shift in a different world, other parts of what make them intrinsically who they are as characters would stay the same, and I write to get to know them, I suppose.
Whether AU or canon-verse, I think I’m writing to get to know the characters more. I want to see how they would act in a world I’m familiar with, like a boarding school, or a fantasy setting, or pirates on the high seas, etc. I want to see how, in canon, it might be possible for characters to work out happy endings for themselves, find hope against the odds. For example, for my HxH fics, I want to examine Hisoka and find out what makes him who he is, but more importantly, what he wants, because if I know what he wants it’s easier to start scraping away at him to try to force him to develop in a way I would like him to (redemptively). Canon is amazing in HxH and I have no complaints really, but it’s still fun to explore the possibilities, because I believe the world is full of possibilities and our choices with whatever we’re given do matter, so I like exploring scenarios in which characters would be driven to realizations and potentially growth.
Characters are like friends. To be clear, I think you absolutely must maintain a healthy detachment to the extent that, if someone says “I don’t like x character” and you think that’s an insult on you and lash out back, you need to step back from fandom. But I also think characters are powerful, powerful creations that can truly impact people, and you can maintain a healthy perspective and still truly love them. Kaneki Ken, for example, saved my life. People can genuinely love characters and suffer with them, and grow with them. I’d like to explore those points of connection with the characters I connect with in my fanfictions. I think it helps me know myself, too. The more I write, the more I know what I think and believe. So I completely relate to what you say about using it as a way to process your feelings!
Writing, for me, whether original or fanfiction, is relaxing and stressful at the same time. It’s hard and it’s lifegiving, and it’s gotten me up in the morning and made me anxious at the same time. It’s fun, or I wouldn’t do it, but it’s also work, and fanfiction’s no less work than original fic, at least not inherently. But it’s a way to be involved with something I love (stories, and fandom).

