The trailer/tail lights aren't related to the interior 12 volt system at
all, except where the tow vehicle charge line connects to the battery
(via an isolator) and the breakaway switch on the tongue. You may have
numerous unrelated problems, but I'd check and make sure the switch
isn't tripped first.
The best way to troubleshoot the 12v wiring is with a test light,
starting at the battery (or converter output), and trace through the
system until you find where it all stops.
Same with the running lights. Make sure you have power at the tow
vehicle plug, then at the splice point (plate on belly skin panel on
late '60's trailers), then on to the individual lights. The usual
culprit is the grounds (as Dr.G noted).
Best of luck,
RJ