Are global temperatures zero-sum, i.e. if we’re experiencing lower-than-average temperatures, must some other area(s) be higher-than-average?
Doubt it, as total solar input changes over time. The sun goes out, and there’s not going to be some spot left on earth with a temp of 8793°K just to match out all the 4.4°K readings.
You may have encountered some variant on a discussion of the Hairy ball theorem
That’s the one that says you can’t comb a billiard ball without a tuft, or have an earth where the wind is blowing at all points:
If you mean “with the current wave of low temperatures in the US, dose this make the Arctic warmer than normal,” I suspect the answer is yes. The wave of cold air flowing down from the Arctic needs to be replaced from somewhere, and that somewhere is going to be warmer than the North Pole.
My guess is yes. The energy hitting the earth from the sun is relatively constant, as is the energy radiating into space from earth.
Could there not also be temporary changes in the Earth’s reflectivity and cloud cover which could increase or decrease the heat temporarily over a couple days or so?
Nope. It was warmer in the Mesozoic and cooler during the ice ages.
Unless it’s being replaced by air at altitude which would be cooler.
And, where is that air being replaced from?
The question has to be asked how that varies related to time.
Solar incidence changes because solar output varies. What is the timescale of variation? Something on the order of hours, or days, or years?
The Earth’s radiative output is based upon Earth’s temperature. If global mean temp is higher, Earth is radiating slightly more. What is the scale of this effect?
Cloud cover, snowcover, soot and ash, etc all affect Earth’s reflectivity. That means more or less energy not captured by Earth. How does this vary over different time scales? Can this vary substantially over the course of a day, or is it more of a yearly average kind of impact?
Over long time periods, we can definitely state no, but I think the OP is more concerned with short time periods, i.e. “instantanous” or within the same day or few hours or something.
Without numbers on the above, everything is a SWAG. My gut says thermal inertia plays a role, and the response of all the features above are longer term effects rather than very short term changes.
Also wondering how air pressure fits in.
Sort of, but not exactly.
What must be higher-than-average is the same place, or some other place, relatively proximately in time. This is what keeps GATA (Global Average Temperature Assessment, I think) well, average.
In the short run this varies very little, and so over a period of a year or two, it all averages out. By definition this means that if it’s extra hot right here, right now, it’s gonna be extra cold right here pretty soon, or extra cold somewhere else right now, orextra cold somewhere else pretty soon.
Now over time (let’s call weather over a longer period of time “climate”) GATA does change; that’s what gets referenced when folks worry about Global Warming: GATA is increasing.
It’s surprisingly tricky to measure GATA. The two places I know of are Goddard Institute for Space Studies and University of East Anglia (yes; that one, of email fame). So of course that’s why external “proofs” of longer-term climate change are offerred, such as polar ice diminution–it turns out you have to do a lot of tricky 
stuff to get a good GATA.
GATA s/b Global Averaged Temperature Anomaly.
Sorry.
Air at higher altitude further south which is actually colder because the tropopause is higher (High altitude air at the equator is colder than high altitude air at the poles.)
And, where is THAT air being replaced from?
Do you see what I’m getting at? Unless the Earth is getting colder on average, one a cold spell in one area implies that something else is warmer than average.
That air is being replaced by rising warm air which cools as its pressure drops.
Yeah I see what you are getting at but I don’t think it works as neatly as that. Cold weather in some part of the USA does not necessarily mean warm weather elsewhere. Specifically I mean that cold surface temperatures in one place does not necessarily mean warm surface temps in some other place.
Take a look at this current weather report: http://www.weather.com/multimedia/videoplayer.html?clip=10827
You will notice that the cold front that is going to freeze the East coast is followed by above-average temperatures as the front passes, which indicates that the cold northern air is being replaced by warmer air as it is blown South.
Yup. The earth’s albedo changes over time. That affects temperature:
The Terra satellite continually monitors changes in the planet’s albedo.