You asked for lanolin questions and I have arrived to fulfill!
What do you believe her childhood was like?
ohohoho you really opened the floodgates with this one >:3c
So we've all seen the ABT post of Lanolin's mother (who I'll use the fandom name of Fleece for since I don't have any better name ideas), it's pretty obvious that Lano was taught to keep up appearances and prioritize her needs below looking good. I think a lot of her early years were spent being taught to be a perfect pretty doll to satisfy her mother's expectations.
That being said, while I DO think that Fleece was frequently nasty and unsympathetic, and that Lanolin learned early on that she couldn't count on her mom for comfort and that she would have to swallow any unpresentable emotions in order to be accepted, I don't think Fleece was doing this purely out of spite and narcissism. Fleece definitely has a narcissistic streak, but I think her motivations are rooted in antiquated ideas of tough love rather than deliberate neglect or outright hatred of her daughter. I think Fleece was probably raised in some dire situations of her own, situations in which being appealing and femininely charismatic and acceptably pretty were a necessity for survival. Situations in which Sonic-style platitudes like "stand up for yourself, just be confident in who you are, live your own life" were not an option.
So, Fleece takes any pushback from Lanolin as a rejection of her worldview and a personal insult. After all, Fleece knows how the world works, she's the adult, so the child is just being obstinate and disrespectful and needs to learn how to be a proper lady. It's for her own good! Manage your own emotions and perform correctly, because that's the only way to secure your safety. While I said Fleece doesn't hate her daughter, I do think there is a touch of resentment in her, like "I had to go through all this and learn all these tough lessons so you do too!"
This, in my opinion, explains Lano's first appearance in the comics quite well. She's being the helpless little girl her mother raised her to be, but even so she cares about her village! She can't fight, but she just watched Sonic and Amy take down the giant crab, so she goes to them to get them to help when the badnik attack continues! This emphasizes that Lanolin DOES care a lot about others, that she has a heroic streak in her, even if she's limited at this point in how she can express it.
I think being raised like this leads to a complicated tug-of-war in Lano's sense of identity and how she wants to live, in which she is both resentful of but still very affected by the demands placed on her by her mother, which would need a whole other long-ass essay for me to fully explain. In this excellent post by @villainboss24 he mentions that Lanolin actually likes cream and sugar in her coffee, rather than black. I personally take it a step further, where she does prefer cream and sugar, but she drinks it black anyway because anything "frivolous" like that reminds her of her mother, with a nasty, tangled combination of rejecting anything that reminds her of enforced femininity as well as not allowing herself to have anything nice.
Getting back to the actual experience of her childhood, I imagine that, due to her mother, it was a lot of running up against constant arbitrary barriers, getting frustrated by them, then being punished for expressing that frustration (the autism headcanon makes a lot of sense). I don't think it was nonstop grimdark misery, she definitely had friends and got to have fun sometimes, even if the pressure in the back of her mind to look and behave correctly was probably omnipresent. If Lanolin really was nothing but a hateful selfish misanthropic bitch like her detractors claim she is, she wouldn't have been inspired to join the resistance to help others, so I think it's fair to say she had enough positive experiences in childhood, enough people she cared about, to start thinking about protecting people and making the world a better place.
Did she have a dad? Idk honestly. Her mother's influence on her is clear, so if her dad was around I think it's a fair enough assumption that he was less involved in Lanolin's upbringing than Fleece was. There are tons of different ways to explore the idea. There's the typical absentee father single mother situation, or perhaps they were both around but then there was a divorce, or maybe her dad died. In fact I saw a post from a couple years ago saying that her mom was actually dead, which is a really interesting idea and stirs up all sorts of fun complicated feelings. I personally like to headcanon that Fleece could have been one of Clutch's floozies back in the bad old days, but that's just one idea.
Thank you so much for this ask omg I could yap about Lanolin's mommy issues forever!!! This is all just my own interpretation obviously and ppl are allowed to disagree with all of my takes here; I don't intend to come across like I think I'm an authority on anything.