There's a lot to unpack here, this is actually several questions.
First of all, the way your circuit is designed currently, your data pins are connected via the transistors to ground. This means if you set a pin high, or use analogWrite(), the Arduino will pump as much current as it can get, directly into the ground. This can damage the Arduino. Put a resistor between the pin, and the transistor.
The first point is that you are using several LEDs. If you want to control them separately, you will need 3 pins per colour. AnalogWrite() only works on PWM pins - if you look at your Uno, you can see that some of the pins are marked with a ~ (pin 3,45,6,9,10,11) - these are the only ones you can (easily) use. Since there are 6 of them, you could only run 2 LEDs this way. You could either run the project on multiple arduinos (1 Arduino per 2 "windows"), or you can get a chip to do the heavy lifting for you. I am working on a similar project, and am using a TLC5940 - there is a thorough tutorial at http://playground.arduino.cc/Learning/TLC5940 - they also save you from having all the resistors and transistors, and each TLC5940 will run up to 16 LED-pins (3 per LED, so you can run up to 5 "windows"). They are also designed to daisy chain, so if you need more, you can just add another chip; no extra Arduino pins to worry about. 3 of them can run all 12 of your LEDs independently.
The next item to be aware of is that there is a limit to how much power (in Amps, or milliAmps) you can draw from a given power source. Each part will use a certain number of millAmps. The spec sheet for your parts should tell you how many. LEDs are generally around 20mA (per colour), but this varies - check the datasheet. An Arduino uses about 50mA. This means that if all 12 of your LEDs are bright white, you will be drawing 50mA+(20mA * 12 * 3) = 770mA.
How does this affect you? Well, if your project draws less than the power source can provide, you're golden! You can't provide "too much current" to a component - it will draw what it needs. When you draw more than the supply can supply, you get one of two results. If it runs on the mains, it will overheat, and eventually either burn out, or catch fire (there are exceptions to this, but you'd better be sure of your facts before risking it!). How long this takes, depends on how much current you are drawing, and how much margin there is on the power supply. In the case of a battery, the battery will drop in voltage, and will also heat up (but is much less likely to catch fire). I suspect this is happening here. If you have a multimeter, you can hook up your circuit to batteries, and then measure the voltage between + and - on the batteries.
P.S - +1 to @Gerben - you need one resistor per LED leg (except common). If you use the TLC5940, you would need 3 TLC5940s; as I said before, you can daisy-chain them. This does, however, mean that you can control each LED individually.