I've already calculated the resistance to be 100 ohm for Red, 167 ohm for Green/Blue when using a 5V power supply.
So why does your schematic show 33 ohms for all 3 LEDs? In any case, you haven't allowed for the forward voltage of the LEDs, or the saturation voltage of the ULN2003A.
The Radioshack 38kHz IR receiver (276-640) says to supply with .56mA max under no signal input. I'm assuming they meant 56mA.
No. they mean 0.56mA.
I=5v/330ohm I =15mA -- Why such a low amperage when IR receiver says to use 56mA?
The purpose of the 330 ohm resistor is not to reduce the voltage, it is to prevent noise on the +5V supply reaching the IR receiver (in conjunction with the 4.7uF capacitor).
Since I'm using a low powered RGB led then I assume I don't need the ULN2003A driver to manage voltage to a high powered 5W bulb. Is that correct? I will wire up D11 to Green, D9 to RED, D6 to Blue.
The absolute maximum output current per Arduino pin is 40mA, which is below the 50mA you want for your red LED. So you do need a buffer, unless you are prepared to run your LEDs at lower current.
I see a 4.7microFarad capacitor is wired in between the Vs input and GND of the IR Receiver. I understand that the capacitor stabilizes the current, but why do I need that for an IR receiver? Is it because detecting wavelengths (nm) accurately requires a stable voltage?
It's because in the applications for which they are normally used (e.g. TV remote control receivers), the 5V supply is normally derived from a standby switching regulator and is likely to have quite a lot of noise on it. Any noise at 38KHz could lead to the IR receiver thinking it was receiving a signal. The 330 ohm resistor and 4.7uF capacitor filter out such noise.