I've generally viewed local government as a reflection of the government of the nation-state as a whole. For example, take medieval Europe. The feudal system allowed a caste of nobles who answered to the king to unilaterally rule over their lands and the people who lived in them. Major cities were the strongholds of the nobles; you could say that they were mayors with additional power, both in the cities and the lands around.
However, if you replaced the monarchy with a republic, then there would be a different system of government in the cities. Now the power is in the hands of the people, and they can rise up. They elect representatives who are against the nobility caste, and the nobles are overthrown, replaced by relatively impotent mayors or, more likely, councils made up of wealthy landowners (in this example medieval society). Alternatively, you could just chuck this system into place form the start, but that assumes a certain political evolution that somehow avoided a monarchy.
Your society is at a medieval technological level (by a coincidence; I wrote this part before seeing the edit), so it makes sense that a peasant/lord society will develop - a system of feudalism. The economic, political, and medieval power rests in the hands of the nobility, and so I would wager that you would have a stereotypical medieval political system. A larger monarchy might develop from warring kingdoms.
The one thing here that might change that is the fact that you have multiple races. If Tolkien is any indication of reality (and in this case, I think he is), this kind of one-case rulers will not gibe well with the different races. Each one will want to rule itself, probably because each race developed independently. This could make for an augmented nobility - more nobles representing different races - which eventually evolves into some sort of council, with nobles representing each race. The transition to a republic is already happening.
I made the assumption above that the races evolved and developed independently of one another. I'll hold this assumption to remain true throughout the rest of the answer. There is nothing on Earth to back it up, but some logic should support it.
You mention a World War. This world is currently at a medieval technology level, so I'm currently imagining a disaster beyond all known proportions setting the world back in time technologically (and economically, culturally, etc.). That would involve massive devastation - hell, they sunk an entire continent! That type of thing is not easily forgiven. If I was to write a faux history book about that war, I would write one passage like this:
Sergeant Heysha would never forget the final mission. As the plane sank low over Valinshek, its engines screaming, the dirge mixed with the sound of thousands - no, millions upon millions upon millions - of anguished souls screaming out for mercy. The explosions were deafening, as the rest of the squadrons continued on with their runs. But Heysha's plane had been hit and hit hard, and it flew low over the edge of the once-peaceful forest, now scarred by fire and acid. The world was spinning - no that was the plane. Or was it? He looked back over his shoulder at Loujik, the land of the snow-elves, and saw a crack in the Earth open up. The screaming seemed to get louder and louder, and the ground rushed up to greet him, as Loujik seemed to disappear under the smoke and clouds and dust and fire.
The blackness swallowed him, and the pain, when he hit the ground, was as deafening as the screams.
That kind of thing - genocide on a massive scale, because destroying a continent will typically kill a lot of people - will not be forgiven easily. I find it hard to believe that there will be peace between the races for some time. Although your story may be set far enough in the future that the hatred has dissipated. A date for the war could help gauge how much still remains.
But races living together in cities? That seems rather tough.
Culture is a tricky thing to define, and it is what always seems to stump me. It encompasses so much - so many years and so many people. I'll give it a go, though.
Religion. This could go in a lot of different directions. I can see monotheism taking root, although there might be some bias in there because of its grip on the world today. The idea of a king bowing to a god and serving that god - and thereby his people, in an attempt to make the populace support him, and be even more subservient - is one that I always find tempting in a society like this.
There needs to be some religion, at the very least. The lives of most will be crappy, and they'll want to believe that there is something good in the future. You really need religion for that to happen.
Art. Art might circle around religion and religious ideas (unless you take aniconism into account). In fact, I would call it a likely development path (although again, there is some bias).
Music. Who doesn't like a good medieval tale? It might not be as silly as the tale of Sir Robin, but it might be story-centered. Story songs are, in general, my favorites in this type of society. They can be used to venerate kings and nobles, too, and their deeds.
You say that magic is generally frowned upon. That's an interesting scenario because I would bet anything that even if, officially, magic should not be used, you can bet the farm that all the leaders will want to use it. Will it give them an advantage in battle? I would assume so. That means that they'll want it, and they'll want to use it better than anyone else. It will all be very hush-hush, though.
Who has the magic, though. Is it one of the species, members of all races? Who in particular gets it - all members of a race, or just some? These may play big roles in inter-race relations. If the elves have magic, then you can bet anything that some gnomes will want them on their side when fighting other gnomes.
That said, there's always the trope of people fearing magic and those who have it. 'Who's the weirdo with the wand?' 'I don't know, but he's a real creep. Don't go near him.' This could happen on an individual level or a race-to-race level.
This is even further complicated by the possibility that magic was used in the past. Was it used by Heysha's superiors to sink Loujik? If so, then the people who knew those who lived there (the snow-elves) will not want to associate with anyone who has this magic. After all, it killed most of (if not an entire) race!
I think this goes along pretty much with the government, but I suppose I can talk about the way the leader(s) behave(s). After all, in his heyday, Hitler behaved differently than present-day Fidel Castro, even though both had/have quite a lot of power.
In this feudal-ish society, I would expect that the rallying cry would be around the king, and it would be emphasized that people have a "duty" to help him, as he is their lord. It might focus on loyalty, and simply that. The peasants have negligible power, so it's not as if they need to be quelled or suppressed (yet!).
I've so far stuck to the feudal system and the associated economic structure, and I see no reason to deviate from it now. We can stick with the peasants/nobles power structure, which comes down to the majority of people being fairly poor and farming or working in the fields for the few rich folk.
The major trade hubs will be cities (of course!), where merchants, artisans, and skilled tradespeople will make their homes. Inter-city trade will fall into the hands of the merchants, as most people will probably not go too far from their homes, being fairly powerless.
The nobles will want to protect themselves, so I anticipate a fair amount of technological development being devoted to the military (hey, that's just like today!). Better crossbows, guns, battering rams, and the like will be needed, and so there will be a thriving market for them. The tradespeople who can make them will gain some economic (and perhaps political) standing.
Whatever you choose, though, make sure that the cities are the trading hubs. There won't be much infrastructure outside of city-city trading routes, so the people will have to go to the cities in order to get goods.
Who has the power in this scenario? The king and the nobles. Therefore, they also hold control of the military. On a large scale, things can get complicated. On a small scale, they're easier to predict.
Each city will likely have:
- A guard positioned around the perimeter. I'm envisioning castle-like fortifications, which means guards on the ground and higher up on the walls. These guards will search any suspicions persons from the outside.
- A pseudo-police force that reports to the noble(s) in charge. They will arrest people that break any laws in place, and possibly go for political enemies, as well.
- A small-scale militia, with the noble(s) at its head. This will be responsible for attacking other cities and defending the city when it is under attack. They may also be deployed to the surrounding lands, but it is also likely that there will be a separate force for that.
The noble(s) will be the ultimate commanders of all of these groups.