1. Why did Belgrade end up such a small city? Did someone nuke Belgrad? Especially as this is supposed to be a Serbia-based state...
Belgrade is, in fact, the largest city in the Balkan Federation, which I mentioned in the description. However, beacuase this is a political map, it makes no distinction between non-capital megacity and a tiny village that happens to also be the administrative centre of that canton. When, last year, the capital of Morava Canton was moved to Kragujevac bracuse of its more central position, maps like these show Belgrade as only another town, when in fact it is the Federation’s largest city and also its financial centre. So no, no one nuked Belgrade.
Also, what do you mean by Serbia-based state? If you mean that the majority of its population are Serbs, you are correct. If you mean that political power is concentrated in the hands of the Serbs, you are not; politics is heavily decentralised and Serbs actually have comparatively little power, which leads to calls for their secession.
2. Why is the Balkan Federation more popular with tourists than remnant Yugoslavia? After all, they have most of the beautiful coast and islands.
The genocidal dictatorship/war zone nature of the remnant Yugoslavia makes it a fair bit less popular.
3. Who leads the Federation? How democratic are these elections?
The federation has a collective executive, the State Council, which makes decisions as a group. The current chair of the Council is Edi Rama. Its elections are some of the most democratic in the world, except in Bosnia and Upper Dalmatia cantons, where they are occasionally marred by violence and the still-ongoing civil war.