I'm a little confused over the ATX power adapter. What is the purpose of it, more than converting sockets? Does it add a voltage regulator or something else needed by the FPGA Arcade?
Primarily it acts as a socket adapter. But it is also contains some electronics which allows you to switch the power supply on and off with a typical power-on push button. Otherwise you would need a bistable power switch which isn't a standard equipment of any modern computer case.
Is it really needed, or can I just use any +5V lead from the PSU?
You can connect directly to the main +5V lead from any ATX power supply. But in that case you will need a switch to control the power supply (e.g. to short power-on signal line to ground to switch on the main voltage).