Questions about a simple LED circuit using the ULN2003A
To answer your questions:


Correct: it is ok to tie directly to tbe base pin, given the internal resistor. The reason one size fits all, is the combined gain of the darlington pair is so high, and that the output pins' current ratings mean any current at the output within specs will be sustained by the internal resistor at the base.
Yes your calculation looks good.
Look at the circuit diagram below which shows how the darlington pairs are implemented. You can leave COM floating.
You can leave the other pins floating. They should have no bearing on your circuit, since as per the diagram below, they are independent (as long as you do not pull COM low).
Perhaps you have not wired it up correctly, or the chip is dead. For the sake of troubleshooting, try connecting COM directly to ground, which would be equivalent to all transistor pairs saturated on. The LED should light. Perhaps the LED's polarity is reversed. Sanity check by connecting pin 1C directly to ground.