My Lavellan is the curious, inquisitive type. This has been reinforced by Solas compliments, he made that part of her even more prominent. Besides that part of her she tends to be more on the reserved side, with snarky comments with those who she feels close to (like Varric [she was attracted to the fact he is a writer, and really liked how he got along with Cole, though he is more of a companion than a full fledged friend], Dorian, Sera [she didn't get along with Sera at the beginning but they share opinions, and got close on that basis. After Solas broke up with her she went to Sera to sit with her on the roof :(]). In my headcannon she has a younger brother, a hunter, so she did her most to save what's left of her clan (and succeeded) for his sake, but she doesn't feel any special obligation towards her clan. Even though she cares much about her people, she always wanted to see world (for my headcannon - that's why she found herself in the conclave as well as ran in during the ritual). I imagine her heartbroken, to be honest I imagine Solas being the most meaningful connection she ever had with another being (seeing she was quite private). She never did seek love to sweep her off her feet but then came that guy who made her feel like her opinions and inquisitiveness aren't a bother, so she fell in love
.
Post game I'd say she won't stop functioning, she'll be depressed but her reserved nature will suppress it, it would only be visible to her friends. She'll become more of 'OK, what is the problem?' straight on then asking questions. She'll work to keep herself distracted, and study when she could.
She won't return to her clan, she'll find it hard after Solas removed her vallaslin, and she would want to learn more of what happened to her People.
If the Inquisition stops existing she would travel the world, visiting Dalish clans and elven ruins.
Oh, and she does not want children. Although I thought it's because I don't want children, I later understood it just REALLY doesn't fit with her story. Children will bound her and won't let her grow. After she discovered she can grow, why would she give that up? Love doesn't mean that much to her.
Maybe only Solas' love.