Regarding your first paragraph. Yes, pin 4 is the +12VDC circuit. When connected, your tow vehicle will charge the trailer battery.
I would not suspect a bad ground since your left brake light was working.
You should have about 12.7 volts for the turn signals, brake lights, running lights etc.
If you have good connections at the truck and trailer, then check the bulbs themselves. Make sure each bulb is in good shape, and use your meter to make sure you have voltage at the light sockets. Make sure your light bulbs do not have corrosion on the terminals. If so, clean the terminal and the bulb and check again.
