Wire peltier device and fan to power supply
I think you have more or less the right idea, absurd as it may sound. Jamming "all the wires into the 12v lines and turn it on" is pretty much what you'll be doing. 

All of the devices are 12V devices, so no problem there with varied supply voltages. You'll drive them all in parallel- all of the 12V positive supply lines tied to the positive PS output, and all of the negative supply lines tied to the negative input.

The power supply is rated for 40 amps, and the ~36 amps you'll be drawing should not be excessive (but it may be too close to the PS rating for reliable, long term survival).

As far as how to actually connect these- there are a lot of options. Terminal strips with spade lugs and jumpers would be my simplistic solution assuming I had the room.