Something just struck me.... some rental places let you start moving in early, but you can't fully move in (stay there) until the start of the lease. What if the 11th was the actual "start of the lease day" but they started moving stuff in on the 7th. Maybe, EG took the kids outdoors, so they wouldn't get in the way of the movers. EG also had to go back at some point to clean up the apartment... If you don't clean your apartment, they deduct the money from your deposit. I am sure she wanted that damn deposit back. It is possible that EG had done something to Lucas during the apartment clean up. Forgive me for saying this she could have stuffed him in a trash bag and took him out with the trash she had cleaned out of the apartment.
Anyway, I heard the dogs didn't pick up on Lucas' scent in or around the house. It could be because he had never stepped foot inside of the house. We don't know how many hours EG had her boys, she might have had them only 2-3 hours before their bio-dad came to pick them up. She could have gone back to the apartment that day to do the move out cleaning... This is why no one saw Lucas after the 11th because he was disappeared the same day.