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Peltier unit are often found in Coldman electric Cooler so it indeed can be used to cool stuff :)
Not much of a problem, this system run on a peltier unit that is used in overclocked computer. so basically you could use an H50 water radiator from corsair and it would pretty much be bolt on. The drawback is that it consume more electricity. But water is not a good heat conductor compared to air, so you better off upgrading the heatsink/fan on the hot side instead.
Vinegar seem to work nicely to help dislodge the shell, but indeed i will try the baking soda next time, see if it work better
Try to find some 2 stroke aluminium engines, compressor have those with Teflon ring. They should work fine and the cylinder wouldn't rust as fast. If you have access to a machine shop and metal, machining one in stainless steel would solve the corrosion problem, as for the bearing, that another problem. 4 stroke engine would have way less problem with bearing deterioration as long as the oil pan is ventilated and the engine is well lubricated.