Lights on a model slot car
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For the polarity protection diode, a 1N4148, 1N914, or 1N4001 are all very common, inexpensive, through-hole diodes that would do just fine. (Their max voltage and current ratings are well above the minimum needed for your requirement.) The first two are physically smaller which may be helpful within the confines of the model.
You want a capacitor rated for a voltage at least a few volts higher than the voltage you are operating at. In this case, 20V would be okay, but 25V is a more common rating.
This is a non-demanding application for the capacitor, so you can ignore other specs.don't need to worry about max temperature, lifespan, or ESR specs. The higher the capacitance, the longer it will power the LED after losing input power. So a 2200uFd cap would keep the LED lit 2.2 times as long after losing input power as a 1000uFd cap.
The output voltage from the capacitor will drop as it discharges discharges after loss of input power, causing the LED to dim correspondingly.
LEDs usually have rated operating voltages somewhere in the range 1.6 to 3 volts, with the actual value depending on the chemical makeup of the LED. Most red LEDs are under 2.0 volts, and most blue or white LEDs are near 3 volts, with green and yellow LEDs having ratings in between. They need a resistor wired in series with them (in your circuit diagram, the 470 ohm resistor) to limit the current when operating off of higher voltages like 12 volts. There are some LEDs that have a built-in resistor for operation at 12 V. If you actual have one of those, it will be dim with the 470 ohm resistor in your circuit.
You should use a resistor with a 1/2 watt or higher power rating.
You may also want to add a resistor on the power input side, to limit the current charging the capacitor on startup. Otherwise, enough current could be diverted to charging the capacitor to slow your initial motor start - not what you want in a race. ;-) 50 ohm might be a reasonable choice for that inrush limiting resistor.