The ground problem is quite simple but very confusing to watch. The blinker feeds one half of a double filament bulb. Normally that half lights up as the middle junction of the two filaments is connected to ground. Remove the ground and the running light feed now is energized. On the other side of the truck the running light filament has a proper ground and so both sides light up.
If it were me, I would install good test grounds on the left side lights and see what happens. You could also remove the left side bulbs and try it.
