Powering LEDs with a computer's power supply
As you stated you have a total of 5Amps.  At 5 Volts thats 25 Watts which is a no-brainer for a typical computer power supply.  Be aware that even though a CPU power supply may have a given rating, say 200W, that doesn't mean you can load the entire 200 W onto a single voltage rail.  Having said that, you won't a problem with 25W on a single 5V rail.

Regarding the use of the same 5v rail to supply for the Arduino, there won't be any problem with that either.  After all, the power supply is designed for the purpose of powering electronics.