When you plug in the AT plugs to the motherboard you always make sure the black wires of each plug are together it's the black to black rule. I hope that helps and good luck with your project

What about the funny metal bits with plastic sleeving? And when I cross the pin for the power on the board headers, does it work as intended or do I get power as soon as I plug in the power supply?

The funny metal bits with plastic sleeving are what you need to short to turn the PSU on. And keep them shorted to keep the PSU on. Normally, you would do this by connecting those metal bits to one side of an AT power switch. When you turn the switch on, it shorts the metal connections, when you flip the switch off, it disconnects them.