Your testing shows a potential issue with the relay. Let's verify your testing.
Unplug the ignition relay.
Measure the voltage across the battery terminal. (This is battery voltage.)
Measure voltage to ground on each W/R wire. You should have battery voltage at each wire. I understand you had the same voltage on each wire as the battery.
Put the key in ACC and measure voltage on the L/R wire. You should have battery voltage. I understand you had the same voltage here as the battery.
Put the key in ON and measure voltage on the L/R wire and the B/W wire. Both should have battery voltage. I understand you had the same voltage on each wire as the battery.
If all those tests pass, turn the key to OFF and plug in the relay.
Put the key in ON and measure voltage at the RADIO fuse and the FUEL GAUGE fuse. Both should have battery voltage. I understand that you had voltage at the RADIO fuse but not the FUEL GAUGE fuse. Furthermore, the turn signals did not work.
The fuel gauge and turn signals are powered by the same side of the relay. If you don't have voltage for the gauges at the fuse box (with a good fuse), I would expect the turn signals to be non-functional, too. I'm also betting that your reverse lights aren't coming on, either.
Let's look at what all powered by the W/B wire.
W/B Wire
Floor Sensor
Cal only
Fuel Pump
Not during cranking
Inhibitor Sw
AT only
Fuse Box (IGN)
G
Turn Signal
Brake Warn Lamp
Turn Signal Power
Seat Belt Warn Tmr
Fuel Lvl Warn Lamp
Reverse Lights
L
Gauges
Cooling Fan Relay
Fuel Gauge
Water Temp Gauge
Oil Temp Gauge
Tachometer
Speedometer
Unk Func
Kickdown Sw
AT only
Among other things if that relay is faulty, you could lose power to your fuel pump when the key is in ON. If the coil or contacts are breaking down, it would not surprise me if it was worse when the car is hot. Your engine could fire with the key in start and die shortly after when the key goes to ON due to the pump losing power.
As far as connecting the relay in the wrong direction, yes it's possible but difficult.
That edge should hit the piece of phenolic that is near the one row of pins on the relay. Also if the relay is attached to the relay panel, I would expect that you would have to twist the connector significantly to get it to be in the wrong orientation.