Calculating the current

Think as power supplys as a constant voltage, rather than the current they can provide. So a constant voltage supply will try to maintain the same voltage independent of the load you put on it (until, as you said, it blows up). So, a fan is designed to "pull" or "let pass" 1A for a given voltage while an air conditioning device is designed to "pull" 10A for the same given voltage. Thats why they pull different currents. And, while you can "force" more current with more voltage, some devices are smart enough that they will try to compensate for that using their regulators (switching or linear) by having their own constant voltage supplies on the inside, thus maintining about the same current consumption up to a given voltage. Normaly supplys fail not because they fail to push the current, but because they fail to provide the current that is being "pulled". If you have a constant current supply, when you try to "push" more current to a given resistive load, the voltage will rise.

About the phone, battery charging ICs will often have a limit to the current they can charge (as will the batteries). Often on cellphones that limit is close to 1A. Hence you can charge it faster on 1500mA. The 200mA rating is probably based on the USB standard max current, and is obviously easier for the manufacturer to supply you with the phone because its cheaper than a 1500mA supply.

p.s.: to better understand different current draws for same voltages: http://en.wikipedia.org/wiki/Ohm%27s_law also remember that not all loads are resistive(most arent)

I'm driving the display with a buck regulator set to 2.2V for testing

Don't do that. An LED is not like a resistor. Once over a certain threshold voltage the current increases rapidly, so small changes in supply voltage and/or circuit resistance will cause a large change in LED current. The graph below shows some examples of LED current vs voltage. Here you can see that increasing the red LED's voltage from 1.7V to 1.85V (a mere 9% increase) caused a ten-fold (1000%) increase in current draw!

You should set the power supply to a higher voltage, and limit the current with a resistor in series. The resistor drops the voltage difference between the power supply and the LED, resulting in a current flow according to Ohm's Law, I = V / R (where V is Vsupply - Vled).

For this calculation You can assume that a red LED drops a constant voltage of ~1.9V, orange/yellow ~2V and green ~2.1V (which isn't quite true since the voltage does increase at higher current, but close enough for most purposes). If you want greater accuracy then you will need to measure the voltage drop of your LEDs at different currents.

Your multimeter may suck, but you shouldn't blame it for showing different current readings on different ranges. Most meters read current by measuring the voltage across a low value shunt resistance. If the voltage required for a full scale reading is the same (eg. 100mV) then the shunt resistance value must must be higher on the lower ranges. Since your circuit is very sensitive to series resistance, even the small resistance of your meter shunt is enough to change the current.

When you inserted a 100Ω resistor and measured the voltage across it, you effectively added a large value shunt resistance. The current then dropped very low due to the small difference between the power supply voltage and voltage drop of the LED. The answer to this problem is to keep the large value resistor in the circuit, and raise the supply voltage until you get a reasonable current draw.