I understand the Apollo 13 CM got down to 39°F after it was powered down following the explosion in the oxygen tank.
Could the interior of the ISS be cooled (for more than 12 hours) to below 32°F while keeping the atmospheric composition and pressure unchanged from standard values, with the equipment already on the station?
In the general case of current and recent human space stations: probably yes. Salyut-7 froze up after a few months without power, and the repair crew had to wear furs (because they were Russians, and so of course they had furs). In the specific case of the ISS... not sure, maybe yes.
The ISS has a fairly hefty liquid cooling system with some big radiators already. It is remotely possible that you could just crank it up, and everything would get cold enough. Finding out whether this is possible or not has proven to be quite difficult, but you might get more information with a carefully worded question on space exploration.SE.
There are lots of moderately-to-very complicated documents out there on parts of the ISS. I had a read of the Active Thermal Control System (ATCS) Overview. Things that jumped out at me:
Internal Active Thermal Control System (IATCS) The ITCS uses water because it is an efficient thermal transport fluid and is safe inside a habitable module.
The LTL [low temperature cooling loop] is designed to operate at 40° F (4° C)
This suggests that you may have problems if you freeze your station (because that cooling loop may freeze too... hopefully it won't burst its pipes!) but also more importantly it suggests that you can't use the ISS's own cooling system to easily freeze itself. The less meatbag friendly ammonia cooling loop can obviously cool down to much lower temperatures, but it is safely segregated from the habitable areas.
This means you'll have to rely more on "natural" cooling below 4 degrees C, and your rate of cooling at and below 0 degrees C is likely to be lower because the active heat transfer mechanisms will have stopped working.
To make the most of your cooling systems (and natural radiation) you'll need to lower the heat load of your station.
The hull of the ISS is very well insulated to keep out solar radiation. From a quick read of this old article on "Staying Cool on the ISS", it is suggested that windows are a source of "heat leakage". So you should probably seal up the windows, ideally from the outside, with some nice shiny insulation. I'm not sure how useful it would be to unseal them again when you're shaded from the sun... heat loss from the windows probably isn't great.
Next, solar panels. I can't get any figures on how much heat the panels develop, and how much of it either a) is conducted into the rest of the station or b) uses up capacity in the heat rejection systems, but it is possible that rotating the solar panels so that they were no longer face-on to the sun would reduce heat loading slightly.
Next: power draw. The ISS has a fairly substantial power system... 120KW or so. Whatever you drive with that is unlikely to be 100% efficient, so turn as much off as you think you can get away with. Obviously the pumps for the heat exchangers need to stay running, and it is entirely possible that there are some other non-optional critical systems that absolutely cannot be deactivated (attitude control, maybe? lifesupport, possibly, but for only 12 hours you might get away with it) but all those science experiments, lighting systems, computers, whatever... turn em off.
Finally, meatbags. An astronaut at rest still has to run their metabolism. An astronaut working hard might develop as much as 500W of power (for short periods) which is going to involve a lot of body heat. Cut that out. Read a book. Don't think too hard.
Will all this be enough? Hard to say, but it seems plausible.
Use a heat pump to pump heat into an extremity until it is hot enough to radiate light. This will happen at the lowest temperature with what are called black bodies, but anything will radiate light once it reaches the Draper point.
Even if it doesn't radiate light, you can still concentrate heat with a heat pump. That will effectively cool most of the spacecraft. The heat pump will become less efficient as the temperature differential increases.
The Earthbound example of this is an air conditioner. It pumps heat out of the interior of a building and exhausts it outside. It relies on the more efficient convection to exhaust heat, but the same principle will work more slowly with pure radiation. And the air conditioner is less effective the hotter it is outside, as its radiator will cool more slowly.
