When doing a Companion's quest where you have to join one of the twins, Lydia leaves your party. The problem now is that I can't find her. I know for a fact that she didn't die, but she's not at Dragonsreach or my house. Is this a bug?

I have had this happen as well. Check around where you last had her she might have remained there. However if you wait long enough she will return to your house. She might even send you a msg that she is tired of waiting and will be returning home.

I checked the Companion's hall and Dustman's Cairn, but I couldn't find her. Hopefully she'll return, but I suspect she's gone forever.

NPC's usually can stay waiting for up to a week game time... If I remember correctly.

She will eventually go back to her starting location.