Most Helpful Girl
It's quite Obvious here, dear, the Ex has not moved on and probably will Not for a long time to come. You both share a daughter and with This, It is Harder to even be in a Real Relationship with a few skeletons in his closet.
He has his own Life with his own business on FB, Which explains a lot about his Past, and doesn't want the newbie on there to take up any of his own Space nor get in his Face. It's a Privacy thing.
As far as the 'Park her car on the other side of the building,' This is for your Benefit so he doesn't have to Explain anything about the Present... Again, a Privacy issue.
It seems he has two Lives.. The Past and the Present. I also feel he still has feelings for you and she could just be a Rebound Rebeca.
Good luck. xx