Family - Ash
To Lavi famiy is something is unattainable for him, something that he'll never get to have and probably won't ever experience. He doesn't even remember his own family, it's just always been him and Bookman and the old man doesn't treat him much like a grandson. So his views on family are just from what he's seen distantly, what he's heard or read about.
He knows they fight and bicker over the stupidest things but they always have that bond between them. And while some members might not pull their weight or do things that are pretty much unforgiveable they still have one another. It's a bond that can't be broken even if they say "that's not my son/daughter" "I don't have a mother/father". That blood bond or even the bond of love in adopted families is something that can't ever be shaken.
When he was younger and hating the training Bookman put him under, I imagine that Lavi often dreamed like little kids do in those situations, of a family member coming to his rescue. They'd take him away from the cranky old man demanding he learn things and memorize them, they'd clean Lavi up and feed him and then finally take him home to where the rest of his family was waiting. A family that loved him despite having been taken away from them and no longer remembered them, because that's what family does. Love you no matter what you've done or what you've gone through. But as the years went on and no rescue arrived the dreams faded until they rested back in his mind near the barrier put in place of his memories before age six.
But in Facility Lavi has given himself the opportunity to build a family, mostly based on the idea that he won't remember anything of Facility when he goes home. And when he gets taken back by the doctors...well...there's a good chance he won't remember the family he had built. While it pains him to think of the pain he'd be putting them into at his amnesiac return...Lavi's attempts to build a family are continually shut down by the people he lets close being terminated. Those from his home world are of course already family in his mind, including Link, Alma, and Miss Galmar, but it has widened to include Axel, Sora, Ishtar, and even Sasuke on some occasions. But he desperately misses those that have made a big impact on him: Sai, Death the Kid, Yosuke, Celena, Gaara, Lenalee. All of them are family but they're all gone.
So for Lavi, he's pretty much given up on the idea of family again. It's something he's not allowed to have obviously, even with his attempts to let people that close. He yearns for it unknowningly, is jealous of those bonds when he sees them, but refuses to let anyone that close now after losing Syaoran. Family is just not for him.
Scars - Kit
Scars tell stories and stories are what he collects, they often tell the truth where one tries to cover them with lies. Which is why he likes them, likes the stories behind them if he's told what actually happened. They're proof that something. Did. Happen. Even if the person swears until they're blue in the face that nothing did.
Lavi used to have very little scarring on his person despite the kinds of things they go through in the experiments. He counted himself lucky - probably how he used all his luck up - and was amazed at the healing abilities of those locked in with him. But despite their abilities there are some that manage to remain and constantly remind the Bookman of his own story and the truth behind them. The whipmarks on his back (as well as several other
scars) from his time with the slavers in the Abyss, cuts Lavi recieved from his second death from Sebastian's hand are faint marks all over his body. Another mark that he considers a scar is the Tease mark just under his left eye that was placed on him by Tyki while he was dead, apparently the Noah has no plans to remove it despite saying he would.
As for the mental ones, there are so many now it's not even funny. Lavi avoids watching his friends and those he knows from being tortured because he knows he can't stand it. His perfect memory is a horrible thing because just one stray thought and he's going back through it as if he's experiencing it all again. Just from being raped alone Lavi developed touch issues and while he's worked through most of them he still refuses to stand any kind that are surprises. He still has nightmares about the entire experience, still sees Allen's figure imposed over the one of the Fourteenth he really went through everything with. It and the experiments broke him down enough that Deak had to come out for a time.
There's also the scars from losing people, those cavities cut out and not really allowed to heal. And while Lavi really does want to get friends and actually enjoy them once knowing that he's letting himself have real friends...he's developing trust issues. The way he sees it is if he lets this person get close and they end up leaving where does that leave him? He's all alone again without someone to lean on when he needs them. Folks from home can only go so far with this because they know in the back of their minds that he's a Bookman and he'll have to leave, but the folks Lavi meets don't know or understand it so he has a greater shot of having true friendship. Going back to losing these sorts of people, why put the effort into it, why put himself into it, especially if they're just...going to get terminated.
Fantasies - Cas [Canon and Facvi]
Canon Lavi hasn't really had any fantasies honestly aside from the few that come to him in dreams. Usually has Lenalee or Miranda or any of the other DGray ladies in something rather skimpy but not something where his imagination has to put a lot of work into. And roses (the whole flower, not the petals) are usually around them on the bed and they're reaching out to him but then he sort of wakes up and forgets what he's even dreamed about. Mostly because Gramps makes him record his dreams on paper and he doesn't want to do that with those dreams.
And he doesn't really allow himself time or the chance to spend the time that he's not doing anything to do something like daydream. Not with the punishments that can be brought down on him, not what he wants.
Facvi...given his time away from home, his... /cough. Current making out with Kanda (on the 13th), and the things that happened to him in Abyss...the boy is rather torn. On one hand he wants to think about doing intimate things with Kanda but makes himself stop once he get up to the two of them being shirtless and kissing somewhere other than above the neck he makes himself stop. He has to otherwise he's swarmed by really bad memories. But he's hoping that Kanda can help him cover those up with better ones eventually. He knows for a fact that he wants to go all the way with Kanda though prefers to be topping just because the whole bad memories again.
And sometimes he has dreams of being with Tyki or one of the other guys he was close to in Fac before they left. A-and Siri.