1.) When she and Lucifer fell headfirst into the pit Lucifer protected her as best he could and suffered because of it. Smoke from his burning essence, or perhaps something far darker than the Lightbringer, infused itself into her hair during that moment giving it not life but an aspect of wind.
2.) She considers herself a sinner since she's a fallen human, though she's a slightly different breed as the first to fall and only known human to fall alive into the pit.
3.) Lilith is not bound by Lucifer's limitation to not harm sinners. Nor is she technically bound to Hell. If she could get to Earth she could walk among the humans without issue, but obviously wouldn't be allowed back into Eden if she found it.
4.) Her feelings for Adam are still complicated. They were literally made to love each other. To find each other desirable. Before millennia of borderline worship gave him a god complex Adam was a better, kinder man and she remembers him clearly. They simply had very different ideas on how their relationship should be. She was his equal. She refused to be his property.
5.) She was tired of being an 'Exception' to Lucifer's hatred of sinners. Tensions were high between them for a long while before she started her campaign to empower sinners, to try and get them to freely choose to make better choices. Hell doesn't have to be Hellish.
Headcanons subject to change based on lore drops:
1.) Lilith can be just as silly as Lucifer. They bring that out in each other in private, and still find joy in playing.
2.) She does her best to be seen as a serious queen in public and is serious about always wanting to improve Hell for her people. She often 'encouraged' Lucifer to focus on his work instead of his hobbies. Wifey bribes and scolding helped him set a disciplined routine that he's lapsed from.
3.) Her silence was not by choice but part of her punishment for trying to organize sinners. Had she spoken a word or acknowledged her family they would be targeted by angels too strong for even Lucifer to resist for long.
4.) Part of her 'deal with Adam' was that her family would not be harmed directly or indirectly. It was the exterminations that gave the sinners the holy metal to bind and torture Lucifer, breaking the deal and ending her sentence. She knew things would be busy after Pentious' speech, and had to pack anyway, so waited to contact Charlie.
5.) Her powers are based on her voice, with a subset of wind, storms, and vibrations. She cannot control sinners or winners, but makes an effort to inspire them to be their true, best selves.