I don’t know about PADI courses, but at least with regard to the question in the title (Where is the coldest commercial diving site?), it is possible to dive (and snorkel) in the Antarctica. Commercial packages, e.g. here and here (the last one being the first package offered in the world, according to their website).
Google maps might be your friend here. Zoom in to a cold region and search for “scuba”. Two suggestions that pops up are:
Norway:
Alaska:
You could try Silfa in Iceland. I haven’t tried it myself but it is supposed to be pretty awesome.
According to this page (and this)there is a dive centre at Nusfjord on the Lofoten Islands of northern Norway – well above the Arctic Circle. I’m pretty sure a dry suit is compulsory there.
EDIT: So I finally found a good-looking map of sea surface temperature and it turned up some surprising things.
I did find a new contender for coldest dive site: Arctic Canada Dive Adventures will take you on a dive trip to Iqaluit Nunavut, where the water temperature is -1°C and the sea ice is 3 feet thick. But they don’t offer instruction there.
The PADI Dive Centre at Scapa Flow in Orkney is the coldest in the UK, and at a latitude of 59 degrees north it has to be a contender.
It would certainly qualify as a cold water dive centre, and as a bonus you can see the German High Seas Fleet that was scuttled there on 21st June 1919.
From their website, they have the following courses:
From the Dive Site Directory, the water temperature and guidance is:
Water temperature: 4°C (39°F) in April to 14°C (57°F) in September
Suit: A drysuit is highly recommended
Credit:stackoverflow.com‘