From Lev's perspective they're leaving the house quite a lot actually!
Lev is the happiest he has ever been. There's something almost decadent about doing normal life things with another person: baking cookies and eating them with someone else; reading a book cuddling with someone else; watching a movie with someone else, and talking with him about it afterward, because you both just had the same experience, because you are doing things together all the time. Lev identifies this feeling as "being in love"; a person who had more experience with it would probably identify it as "not being lonely."
Lev finds out that Sasha has never seen Star Trek, is horrified, and declares that they have to watch every single episode of TOS while Lev explains that Kick and Spock are boyfriends.
They have a fair amount of sex. Lev does not take off his clothes and moves Sasha's hands away when Sasha tries to touch him. About half the time, he jerks off in the bathroom afterward; about half the time he stays cuddled up to Sasha.
This is not a long-term sustainable solution, but Lev isn't planning on doing this long-term. He doesn't want to see disgust flash across Sasha's face if Lev takes off his shirt; he doesn't want to look at Sasha and know that he's bored by giving Lev pleasure. It's harm reduction, in a way. Sasha might not really want to have sex with him, but normal people wouldn't really mind having someone totally focused on giving them lots of orgasms even if that person was needy and unattractive. If he just only focuses on Sasha, then Sasha won't be fucked up about it, and Lev won't be a horrible person.
He is uncomfortably aware that this is a rationalization.
Sometimes, when they're cuddling, Lev disappears for ten minutes and sits in the bathroom and thinks about how Sasha doesn't love him. But it's okay. He knows exactly what to do to cover up that he's been crying.