For the most part, it sounds as though your ground is dead. Trace the ground wire back to where it is connected to the frame of the trailer. You'll probably find that it is corroded where it's connected and it isn't getting a proper connection. It should be the green/brown wire on the right hand side in the second picture. Disconnect the ground at the frame and clean up the area with a wire brush or some sand paper. You'll also want to clean up the wire connector at the same time.
A powered testing source (something which will light the lamps) might be of great assistance in helping you check the rest of the connections. You could also use a 12vdc battery source and just connect it up right to test the connections.
The connector looks very corroded. Try some PB Blaster on the connection. Also, I've found once the connection is together between the vehicle and the trailer, twisting the two against each other (back and forth a little) will produce a better connection.
If this fails, you also need to check the vehicles connection which may also have a fault. You'd need to have someone help you with this, though.