I was thinking of buying 20 or 40 of these, and wiring them in series. I would then sandwich the whole assembly between two metal plates with heatsinks on both sides.
As long as the "metal" is aluminum, copper or silver and absolutely plane, you have a good start.
However... If you connect them to the mains, you have a less good ending.
The cool side would be inside, and actively cooled by fans.
Heated if anything. To cool the air, you need as much of it to pass the heatsink fins of the Peltiers in a given time frame - they are heat pumps, moving the heat from one side to the other (and v.v. if you reverse the polarity).
To power the whole thing, I would run 120VAC power through diodes and a big capacitor, to hopefully provide something close to 120VDC.
The mains voltage is AC, but expressed by the RMS value. When you peak rectify into a capacitor, you get the peak voltage of the waveform, i.e. Sqr(2)*RMS = 1.41*120 = 170V (for a perfect sine, but very close to anyway).
Further, the nominal mains voltage can vary enormously +/-10% in the US IIRC and add to that the effect of returning from brownouts and load dump transients - mains voltage can be an ugly experience, even if you don't touch it.
I was thinking of buying 20 or 40 of these, and wiring them in series [...]
The reason to run them in parallel [...]
Err... Hard to decide?
would be to maximize their efficiency by running them at a lower voltage than the 12V they are rated at, and reduce the amount of heat dissipated by each individual peltier (thus reducing the risk of overheating). I was also hoping to avoid having to find a transformer that can handle a couple hundred watts, thus the idea of trying to convert 120VAC directly into DC. One problem I have though, is that I don't know where I could find heatsinks with very large surface areas, suitable for this kind of purpose.
The sinks you'd have to make yourself - 2mm aluminum plate and a bending brake should help. Make them different widths, so they can stack with 8..10mm of air between fins and bolt them all down to a 10..20mm thick aluminum plate to mount the peltiers onto. Perhaps the best would be to make narrow sections, to make sure each Peltier is in best thermal contact on both sides.
Peltiers are better driven from a current source and remember, Peltiers generate heat, so they have to move that from the cold side as well (i.e. you get less efficiency than what it seems at first sight).
Do use low voltage!
And only do this if you want to do it for the experience. If just for keeping from melting, you can get a cheaper (and instant working) heat exchanger/AC cheaper (and with a higher efficiency) than what you'll have to pay for Peltiers, aluminum, transformers, controller etc. and let's not forget all the hours that you'll use building it (and the heat you'll generate from that work
You'd have to make a hole in an outside wall, the size of the entire Peltier sandwich and insulate properly between the two heatsinks and around the entire sandwich.
You'd also need to make the Peltiers able to breathe to the outside (or seal them 100% against moisture, which is as good as impossible without hurting efficiency even further), as they tend to collect lots of water out of the air - water which you don't want inside the house, or trapped in the sandwich corroding things and creating an environment for mildew.
Leaning back with an ample supply of Scotch on the Rocks is another way to deal with extensive heat
What all this have to do with robotics is beyond me though - a board about home improvement and A/C would IMO be a much better place to ask such questions.