I'd like to think my Lavellan moves on if Solas doesn't come back ever. The Inquisitor isn't going to cry forever and lay down and die; she has work to do that's much more important than relationships.
If she did move on, I have to say that I once liked the idea of Cullen but now I'm picturing her with anyone but him. Just because it feels like everyone has this headcanon about Lavellan running into Cullen's arms. Some were even upset that they couldn't do just that after the ending of Solas' romance. It's left a bit of a sour taste in my mouth! Cullen deserves better than to be a rebound so I'd give him an Inquisitor all his own.
I actually can't imagine Lavellan getting with anyone else in the Inquisition. I'd say Abelas kind of works but I don't know how much he'd want to do with the Inquisitor. Not that we even know where the heck he got off too. And I feel like jumping in with another grumpy ancient elf right after Solas seems strange! Though I guess maybe Lavellan could have a type. 
As for happy endings, I really still can't picture Solas' romance ending happily. I just hope we don't have to kill him in the next game or something. If they do a 'kill him or kill someone else' thing, I'm killing the other person no matter who it is. Solas 4eva no matter what, I'm not blinded by love for him shut up!
I can't imagine my Lavellan ever being interested in Cullen. The Dalish disown elves that mate (for lack of a better word) with other races, and even if Arlaise is not the Dalish's biggest fan right now, I think she's probably internalized that other races = not for romance a long time ago. I know that's not the prevailing opinion, and I don't think it's the "right" or "canon" one, but that's how Arlaise would roll.
As to whether Lavellan moves on or not. I don't know. In real life, it's always healthier for a woman to move on from a relationship that doesn't work. But this is a fantasy world. I imagine that Lavellan senses something special about Solas even though she doesn't know his true identity. She may find that no one else can compare to him or evoke the same emotions in her that he evoked. She may feel that it's better to have no romantic relationship than to settle for someone who isn't Solas. That's how I imagine my Lavellan to be. No real life man is worth all this trouble, but a god is. =)
Haha, this is how I feel. Arlaise doesn't make healthy decisions - she runs headfirst into danger! She fights dragons! She jumps from the third story into the study! She is friends with a Spirit! What normal person is going to match up to that lifestyle? Not to mention Solas helped her through the most difficult part of her life thus far. Anyone new isn't going to share that with her, and it's going to be hard to get past the "Herald of Andraste" title.
Imagine the **** that would go down if suddenly the elves in the alienage stopped aging naturally. Presumably elves in Arlathan still died from accidents and the occasional disease, but all of a sudden Thedas would be facing an impoverished class that isn't going anywhere. Not a lot would happen at first, but eventually, the elves would start to outpace the humans.
I would love to see where that would take us. Would there be riots? Vengeance? Would some elves not want to be immortal? Drama!
How does Thedas compensate for such a dramatic change in what is essentially their worker class?
I bet the prevailing cause of death for City Elves is murder, or something similar. I doubt many of them get to live long enough to die of old age, what with the frequent purging and what not. They'd probably notice that they weren't getting sick first (ancient elves didn't get sick), and then maybe once one of them didn't die they'd be like hrm... It could end poorly for them, if the humans figured it out and fear motivated them to just massacre them. The way they're scattered and generally weak, it wouldn't go well for them unless that immortality meant that they were also all suddenly mages.