You don't want to clean a copper chiller so it is shiny - if you remove the dull color (stable oxide), the metal is more likely to react and form the toxic blue-green oxide (verdigris).
Copper is relatively inert to both wort and beer. With regular use, it
will build up ...
PBW was originally designed to clean a copper "waterfall" chiller device at Coors. It is more safe than caustics and other harsher chemicals both for the copper and you. Also I believe starsan denatures at 80° or 85°C.
From these documents: PDF1 PDF2
Store in cool conditions, away from direct sunlight
Keep containers sealed when not in use
Maximum storage temperature - 30°C
Recommended storage temperature - 10 to 15°C
Minimum storage temperature - Not applicable
The shelf life at the recommended storage temperature is 2 years from date of manufacture
Increasing the ...
As long as it isn't rusted and/or leaking, you're probably fine. If it were me, I'd take some polish to it and make sure to rinse it thoroughly afterwards just to be certain. You may not have to, but my perfectionism would scream at me for any mistakes in the flavor of the beer, even if it had nothing to do with the chiller.
I vividly remember having to ...
Based on what John Palmer says about cleaning copper with various acids, I wouldn't use copper in the acidic environment of fermenting beer.
If you can get your hands on a stainless coil though, the idea will work. SS Brewtech has a Fermentation Temp Stabilization System along these lines.
I use PBW and it is a good 'soak' cleaning agent. However I find when using hard water it can slowly form a 'film' on the surface of things that are repeatedly cleaned by only soaking and rinsing. Any film formed is usually easy to clean off by scrubbing in hot water and re-rinsing. It might be worth checking for any build up in pipes or difficult to reach ...
Rather than circulate a coolant, you could also mount the peltier on top of a long heatsink (SS304) that extends into the beer. Use a H-bridge to change the direction of the peltier so you can switch it from heating or cooling.