I think one reason you are not understanding what is happening is because the controlled lamps are not getting plain old continuous DC power. The controlled lamps are supplied +18VDC that is pulsed in the lamp matrix design. It is better troubleshooted with an oscilloscope. It's current does not change direction, but it does have an AC component. Plus, LEDs are solid state devices that behave differently than a filament lamp in this type of fast pulsing circuit, and there are many types of construction of LEDs with different components internally.

Your statement about the voltage drop being exceeded was not correct, you must exceed the forward voltage to turn on the LED, but as you exceed the forward voltage, the LED resistance drops sharply, current goes up, and an some cases burns out the LED. What you are adding to the LED is called a current limiting resistor.

No, not as high as 18VDC. Varies from game to game, a value usually somewhere between 6 and 12VDC is what the lamp effectively "sees".

http://www.candlepowerforums.com/vb/showthread.php?451101-Multimeter-and-pulsed-DC-current-(PWM)

Sounds fair enough. They're is always LED OCD as an off the shelf option if DIY doesn't cut it. It also has the benefits of controlled fading - which you lose by just sticking LEDs in there.