Sounds like Lime might be male. Best way to tell is when he shreds the paper. . .does he put it under his tail feathers or just let it drop? Only the females put the nesting material under their feathers.

As far as leaving him on his own when you go away, yes, they do miss us a lot because we have become their substitute companion. That is why all my birds have a companion that they share their cage with. But that's not always possible and you have to decide what is best for both you and Lime. If he is approachable and seems sweet-natured then it might be nice for you to be able to handle him. Another option is to get a hand-reared, young lovebird who you can handle and then he will learn to interact with you from the new baby. If you decide to work with him alone and not get another lovebird, then let him have his mirror and leave some music on for him if you are going to be gone for long periods of time during the day. When you go away on vacation, see if you can find a pet shop or pet boarding place that specializes in birds so that he will have other birds there with him while you are gone.

Keep me posted on what you decide.

Sending love,
Cathy

P.S. Teacake and Edla say "Screeeee!"