SO WHAT’S UP WITH LUCIFER?
First off, his motivations. He envisions an existence where there are no humans, no demons, and no God. He wants angels to live and be who and what they are, he wants them to have all their emotions and desires in tact. And after that, he wants to simply let things be. He’s not naive; he knows there will eventually be some angels who rise up and cause trouble, but he’s all right with that. As long as no one will tell them it’s right to give up who they are.
One thing that comes and goes is the desire to destroy the earth--the world itself, not just the people in it. He doesn’t care one way or the other for animals, but I write him sometimes going back and forth on wanting to just completely destroy the planet. Part of this, maybe, is because I don’t think he can and so it's almost like a fantasy for him. If he could just snap his fingers and explode the earth, he would have...that and he’d have no need for Pestilence, Famine, and War. So yeah. Evidence that he can't destroy our planet, for whatever reason.
The other part is I think he likes the earth well enough. It’s not that bad, it was one of the first things created, all in all it's better than Hell. But it reminds him of Daddy and so he wants to smash it to pieces. Plus, it’s not home, but he can’t ever go home, and he resents that.
The curious thing about his homesickness is that, yes, he does miss Heaven. But if given the chance to go back without having to give up what he’s doing, he wouldn’t do it. It’s not home anymore; it’s changed, and it will never be like it was. ‘You can’t go home again’; Lucifer didn’t say that but he may have nodded in agreement with the man who did.
Family is another matter. It’s clear in canon that if he had the chance, he would welcome his siblings with open arms. This hints at a deep capacity for forgiveness that I’m never sure how to play (because I’m not sure how real it is). I don’t think he would have stabbed Michael or Gabriel in the back if they had gone over to his side, and honestly I can’t see him even holding their loyalty to God against them. I think he’d admire it and be touched that they were willing to trust Lucifer enough to leave it behind. It conflicts with the stereotypical portrayal of Satan, though, which is why I love Supernatural’s version so much, and why I second-guess myself so often when I play him.
Oh, but don’t confuse forgiveness with mercy. He has no mercy for people who aren’t on his side, even if he loves them (see: Gabriel, Castiel).
So what’s going on with his family in Adstringendum?
Rhode first. Ah, Rhode...when Lucifer first met her, he thought she was just a little demon. Since he doesn’t have many of those to order around, he entertained her questions. By the middle of their conversation, when he revealed just a part of what he really is, it became clear that Rhode is something else.
Like her, by the end of an hour or two, he was convinced she was a long-lost sibling. In her dimension, perhaps Noah are just fallen angels. He doesn’t care about semantics, what matters is the feeling, the certainty that she belongs with him. And what’s even better? She wants to be with him. Finally, he has a sister on his side, who wasn’t warped and changed by Hell! For the first time since the War, Lucifer had family, loyal family he could actually talk to and be around.
And that brings us to Tyki. Lucifer is used to family hating him, and so he’s expecting Joyd to resent his presence as much as Tyki seems to. As for Tyki himself...Lucifer views him as a sort of pet for Rhode, or at most as a parallel to Sam Winchester: something to be tolerated and protected until the right time comes. He knows that indifference and condescension is obvious to Tyki, but he also knows that hiding it wouldn’t really make Tyki any happier with him, so he doesn’t bother. Besides, pretending otherwise would be a lie. Joyd is family, and Lucifer does not lie to his siblings.
Now Gabriel...that is a sticky situation. Lucifer did not want to kill him. He had to, he would do it again, but he absolutely does not want to. Lucifer’s been in Adstring long enough to know how some things work, and he--stupidly, he knows--hopes that Gabriel will realize this can be like a quiet break, if not a second chance. He would leave Gabriel alone so long as the other angel doesn’t try to interfere with his plans, but honestly? He doesn’t want to do that either. He wants things to be like they were, and he knows they can’t, but he’s missed Gabriel so much...
Every now and then he entertains the thought of them sharing jokes like they once had.
He has more hope for a relationship with Gabriel than he does for Anna. (A lot of this is headcanon but since I doubt we’ll ever get canon clarification I just have to go with it). She’s his younger sister, and he doubts she remembers him as clearly as he remembers her--it was long ago and he had been sort of wrapped up in the War. She grew up very much like he thought she would, so there’s a lot of overjoyed-proud-big-brother going on, but it saddens him because she’s still so devoted to a broken system. She rebelled, but he sees it as a wasted effort because she doesn’t see the big picture. (Of course the ‘big picture’ could also be called ‘Lucifer’s skewed vision of the world’.)
He has a million plans for Adstringendum and the residents, and he has family to help him with it--but now he also has two siblings whom he loves dearly and he wouldn’t mind just being with them with no ulterior motives for a night. But, of course, he can’t because he knows they’ll eventually get in his way.