I believe you're right, that the VD lie was after Lucas went missing.
I need to find the link, but I believe EG herself first told LE that Lucas hadn't seen JO since the incident at the school in Dec ... ???? Then she changes her story...?
I also remember toward the end of the last thread or beginning of this one, a poster put forth several ideas, one of which painted EG as a "hamster" desperately lying to get out of this mess -- PB agreed with that characterization.
EG couldn't account for what happened on VD or why she didn't have Lucas -- so she says JO had him. She couldn't say JO kept him since JH face-timed with Lucas on the 15th.
I have wondered about the sick days when JH was home. I'm not saying Lucas wasn't truly sick, but just a thought -- one of my own kiddos used to think she was throwing up when she spit out anything. If it came back out of her mouth, she thought she was throwing up and would go to the nurse's office thinking she was sick. Could Lucas have wanted to be home with JH--lonely for him, wanting his attention, possibly even hoping for opportunities for 1-1 time with him -- knowing he would hit the road again soon? Or (I hate this), could JH have done anything to hurt him? Could this be why he's steadfastly protecting EG?
The most simple reason why NOT to take Lucas to OG would be fear of questioning about the bruises. But I also think she could have done this before. It could have been to get some time away -- it could have been a punishment for him.