A couple of points:
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The chip on an Arduino is an ATmega168/328, and they like 5.5V or less.
If you are using the PSU in your PC, you may have a spare 5V, so you don't need a voltage regulator (in fact you shouldn't put a 5V regulator on a 5V input voltage, it won't work), but do use capacitors to filter out electrical noise.
You could use a voltage regulator to reduce 12V to 5V. In this case, I'd recommend using an over-specified one (i.e. 2x maximum current) and/or a small heatsink as you'll be converting power to heat in it (heat = power = voltage x current = 7V x Arduino+LEDs) -
There are 12V LEDs, they have a resistor built in:
http://www.rapidonline.com/Electronic-Components/Optoelectronics/5mm-LEDs/5-and-12V-5mm-LED/60113
These cost significantly more than ordinary LEDs and resistors so I avoid them, but they take less space. You can buy 5V LEDs, which would change the whole question. -
In your post "Using in a PC case, Powering more than 5v of LEDs" you said you wanted to drive RGB LEDs, which are different animals again, and are not 12V LEDs. Any component colour of an RGB LED can be driven by 5V.
I'll try to make time to post an answer to your question there, but the summary is:
To drive an RGB LED through all available colours, it needs 3 PWM signals. If you want all 10 of your RGB LEDs to be the same colour, you only need those 3 PWMs current-amplified to drive all of the LEDs. You can drive all the RGB LEDs from 5volts, or a higher voltage, using Darlington transistors or FETs. -
I'd recommend you build the circuit on veroboard/stripboard before designing and making a PCB. Making a PCB is time consuming, can be relatively costly (more costly than 10 RGB LEDs and an ATmega), and can be awkward to change if you get it wrong.
You do need a PCB if the components are Surface Mount (SMD), but AFAIK, none of the things you want are SMD. So IMHO, try to get the electronics right on veroboard. It may be sufficient to the job, and you never need a PCB.
HTH
GB-)