So many questions!!
1) You are right; the current rating is how much the battery is CAPABLE of supplying. It will only supply as much as the load requires. The answer to this this question is summed up in Ohm's Law : V=IxR.
2) You were fortunate you didn't damage the LED. LEDs (and diodes in general) have a "forward voltage", that is, when they are conducting, the voltage across the device remains relatively constant, regardless of the current through them. That's why usually a resistor is placed in series with the LED to limit the current. In your case, your LED forward voltage was close to the supply (3V) voltage, so you didn't pass excessive current through the LED. If you had used, say 9V, you would be asking why your LED was damaged.
3) I have never heard the term "reference current" for motors. Maybe that's a problem in translation. Your motor will draw higher current at higher voltage. But if it's specified as a 5Volt motor, it shouldn't be damaged hooking it up to 5 volts.
4) Again, the battery will essentially maintain a constant voltage and will provide whatever current the load demands. Look at Ohm's law again.
Here's a thought experiment. My house gets 120 Volts from the power company, 200 Amps maximum. If I only plug in a 60 Watt light, I don't pass 200 amps through it, I pass 0.5 Amps.