>What's the reason for not having a common cathode, instead having +/- for each channel? Can the cathode go to a common ground in my circuit, while each of the anodes go to the emitter of each Q1?

But the cathode is common (assuming the led resistor is connected to the anode side), all cathodes are connected to the ground so they are connected to each other too. 
If you have connected the resistors to the cathode side of the leds then you can't connect them together. 
I'm not sure why you mention the emitter, the leds are connected between the collector and ground

>Would the schematic above be sufficient to drive these LEDs? (remember the schematic represents a single channel)

If you increase the base current to be sufficient then it can drive higher power leds too. 

>How do I calculate the resistor values for each channel for a 12VDC power supply? Is the 350mA current for all 3 channels or is it per channel?

The base current should be about 1/10 to 1/20 of the output current to saturate the transistor and have a low voltage drop across the emitter-collector.

When you are using one color (the others are off), you can use 350mA for sure but I'm in doubt of the total max current when all three are on.

>This doesn't seem right to me at all, because the total power comes to 4.4 Watts.

There is nothing wrong in the calculation. What you get is the dissipated power in each resistor for the given input voltage/output current.