The city and the country both have positives and negatives in pollution production... the data here suggests perhaps the urban core produces slightly less pollution (depending upon how you score follow-on effects)...
But both areas fall far far short of the exurbs, where people combine the negatives of both the city and the country for increased comfort and wealth.
We can get a great look at the issue by looking at the US... particularly the CoolClimate Maps from UC-Berkeley. Hopefully it's all the data detail you're hoping for and more... with not just a map, but a breakdown of the different contributions for local transportation, housing, food, goods, and services needs.
However, looking at that, it appears the actual answer to your question is probably it depends, but there's a third party area that outweighs them both...
To take a greater look I selected Oklahoma since it shows a reasonably fair mix of some important features (and I have a large connection to the region). But you'll see the same thing almost anywhere when comparing it to a map. The image shows the overall carbon footprint per household:
You can see that the pollution level within the three largest cities themselves (which account for over 1/4 of the state's population just within those three city limits) is actually quite low.
Likewise though, the very distant rural areas -- large distances from the big cities -- tend to also avoid the darkest oranges and reds primarily.
Much of it indeed goes back to all of your hypotheses... yes, urban areas likely have to have food and such travel further to reach them. But the rural folks also counteract that to varying degrees... by having to travel much greater distances for much smaller resource scales. Whether it's visiting friends/family, having the mail delivered, seeking entertainment, following up medical needs, or many other things, it takes more gas. This image (attributed to CoolClimate Berkeley, but found on Business Insider) shows the additional travel done when living out in the country:
In addition, economics generally causes large bulks of resources to be moved in the most efficient way possible in urban areas, whereas often folks will travel many many miles for small objects that can be obtained more easily in cities.
Plus resources like power and water require more pollution to transport over larger distances or generate locally than they do in the more streamlined bigger cities.
In rural areas, there is some important variability in these maps.
I believe a lot of it comes back to a very central choice made in the calculation methodology: what part of rural pollution for travel and goods should actually be attributed to the urban areas?
Why do I say that? Because it seems the yellower and oranger rural areas line up a great deal with the following map (from the USDA Census of Agriculture):
It appears that greater agricultural production hurts regions. Western Oklahoma (you can see the border on the far left side of my screenshot) is a big wheat area and shows up yellow, as does much of the corn belt nearer the Great Lakes on the larger map (on the website)... while the lighter areas in southeast Oklahoma grow much less... and if you look for greenest colors overall on the larger US map... much of the western half stands out like a "green thumb"... not-so-coincidentally where very little is grown in the deserts sand mountains. (If you look carefully, you can probably pick out the Central Valley of California as a duller green overall, despite its high efficiency)
What causes the agricultural areas to show higher pollution? Probably because they must run significant machinery through the year, and have greater transportation requirements to get in supplies and then deliver their harvest.
So the question then goes to: how much are the resource these agricultural areas using properly being broken down between the rural producers and the urban consumers? In the end, I know not. But it's certainly worth considering. In the end, it might just as well be fair to consider the agricultural resource needs to be more of a sunk cost, as someone must produce the food regardless... and to henceforth instead spread it around equally to all consumers rather than laying it to the fault of the agricultural regions.
In the end, though, the most troubled areas always appear to fall in the same locations: suburban communities. See this map of Atlanta:
On the left is the pollution image. On the right is the loosely matched up city limits of all the larger-sized suburbs around the Atlanta area. And the result again is that not only is downtown Atlanta (the star) not too bad... but even many of the nearby suburbs aren't too bad.
However, near/beyond the black ring (which is about 25 miles), you hit the worst. Atlanta has a lot of sprawl. But I checked dozens of decent-sized cities all around the US, every one showed the same structure, with a greener inner-core, and then a redder area around the edge of the city.
People in such distant suburbs go there seeking to take advantage of the best mashup of the economic benefits of a big city yet with the cheaper costs and rural feel of living nearer the county.
Instead, they end up merging up the greater consumption and larger food import distances of the big city with the greater required travel distances for wants and needs plus decreased efficiency. And the combination ends up being the worst of both worlds, at least in terms of carbon dioxide production.