No to the last one, anonymous. Rather, our regions are merely fragments of larger countries. For example, Kanto, Johto, Hoenn, and Sinnoh are all regions within the country of Japan. These aren’t the only regions in Japan, but they’re the most populous, to be honest. Kalos, meanwhile, is a good portion of France, while Alola and Unova are both vastly different parts of the United States. There are, of course, more regions than this, but these are simply the only ones participating in the National Dex Program at the moment.
As for how big a city is (I assume you mean one of our cities—pokémon do sometimes organize themselves into primitive social structures, but they’re rarely organized enough to construct entire cities), that also depends. For example, Cerulean City isn’t that large at all. Rather, it only has a population of about 140,000 people, most of which are concentrated within a small urban center surrounding the Cerulean Gym. On the other hand, Goldenrod City, the largest city in Johto, hosts about 2 million inhabitants and sprawls across almost all of Goldenrod Bay’s coastline. Goldenrod isn’t even the largest or most populous city among all seven regions of the National Dex Program (it’s the fifth largest, behind Castelia, Mauville, Nimbasa, and Lumiose), but this should give you a rough idea of how large (and small) a city can be.
Finally, with regards to the size of a region … that’s a bit more difficult to say, anonymous. Some regions, such as Johto, are actually rather small. (Johto and Kanto combined only cover about 40000 square miles. It’s one of the reasons why I have no trouble traveling back and forth across both of them as their regional system administrator.) By contrast, other regions can be quite large—such as Kalos, which takes up over 213 thousand square miles.
In short, our world can be quite diverse in terms of both regional size and population, but no, regions are not countries in themselves.
The Pokédex According to Bill 