Regarding the Trooper key in Joseph's pocket, per the picture shown it appears there is some sort of 'keychain' and it may have somehow been broken off. Being that it is in his pocket may not be so mysterious. From Mcgyver's testimony, the Trooper was Summer's and the dodge truck was Joseph's. It's not uncommon for someone to have a copy to their spouse's vehicle. Summer probably had use of the Trooper (hence the children's car seats) AND there were items in the Trooper that she purchased at Ross. So it's quite possible Chase used Summer's key to drive the Trooper. So far as we know, her keys, phone and purse have not been found.