How to connect together: Raspberry, Arduino, PC PSU, Usb Hub, Usb Hdd - without issues?
Seems reasonable at a glance, couple of thoughts:


All the pins marked "GND" on the connectors will be connected together, so can be treated as a common ground. An easy way to test this kind of thing is by using a multimeter set to continuity mode (or ohms mode) and probe between the wires.   
I don't use Arduino, but I thought the DC in needed higher than 5V. According to this page, it needs between 7-12V. This makes sense if it's using a 5V linear regulator as it needs some headroom to regulate correctly. You may need to use the 12V rail, although lower is better (e.g. between 7V and 9V) if you plan on using a lot of current with the Arduino, as the on board regulator dissipates more power and gets hotter the higher the input voltage.
EDIT - to clarify about input voltage:
Looking at the schematic, the Arduino UNO uses a 5V NCP1117 linear regulator at it's DC jack input. From the datasheet we can see the dropout voltage is around 1V (hence the absolute minimum 6V input given in the link above). Supplying the input with 5V will produce an output of ~3.3V (1V + D1 diode drop) - the ATMega will probably still work at this voltage, but the 5V rail will no longer be a regulated 5V rail, so it's obviously not ideal.
Yes, you can use the 12V rail to power the LEDs (assuming current limiting will be used correctly) The number will be limited by the current sourcing capabilities of the rail which should be in the supply specs. Up to 9 small RGB LEDS should be no problem though.