What you're saying sounds awesome. Like, really, really awesome. I think I love Belarus.
Don't get overexcited. It is still unclear whether pros outweigh cons, and whether this system is sustainable in the long run. It is just an alternative approach to development comparing to other Eastern European countries. People prioritize social stability above political pluralism and fast money.
Did it revert to a Brezhnev-style socialist planned economy, or was a new system introduced?
It is a new socialist breed that evolved without demolishing the old foundation. Unlike USSR Belarus cannot ignore the rest of the world, financial markets, globalizing technologies.
I'll put it this way: this new breed takes advantage of the fact that there is capitalism somewhere out there, and feeds of it. For an inside observer - Belarus looks like a socialist society. For an outside observer - Belarus is one united business corporation under single management. But that's what other modern corporations are trying to achieve.
Are there still Pyatiletkas? Is there an equivalent of Gosplan? If planning still exists, in which way did it develop, or is it still done the old Soviet way?
I would again compare it to medium-term planning inside big corporations. There is no Gosplan, plans are developed by the Cabinet of Ministers.
Are prices fixed? To what extent do market forces have an impact on the economy?
It's a combination. Prices are fixed for state produced goods, but since they make a big share of the market they inevitably effect prices of private made goods.
Potentially, market forces have significant impact, but Central Bank learned to navigate things like inflation, exchange rates, unemployment, etc.
Is [private sector] growing or shrinking? Does the bourgeoisie have political power in Belarus?
It mights be growing, but not faster than general wealth of population grows. You cannot become an oligarch overnight even theoretically. Wealthy people have no political power.
Political regime is authoritarian, only the president has all political powers. He is more like system's guardian.
Do Belarussians perceive their country as socialist?
They are not overwhelmed with definitions as long as things stay as they are. People are generally favorable towards Scandinavian / German approaches to the economy, but they are not trying to replicate them.
Wouldn't this make Belarus the most advanced socialist country ever to have existed?
I'm not sure. China and Vietnam are another examples of how you evolve without demolition of the socialist base. Only time will show.
Do the soviets (I mean the councils) still exist? Are there multiple parties? What about the Communist Party? Did Belarus switch to parliamentary democracy as the rest of the USSR did?
There is a Parliament, but it is controlled by the president. Opposition parties (including Communists) exist, but they have small public support and are not present in the Parliament. Soviets exist as local councils.
It is weird, but words "Communists/Communism" have a negative connotation with the Belarusian public while "Soviet/socialist" are rather positive.
what about unemployment? Does it exist? Did it exist in the Soviet Union prior to/during Gorbachev's rule? Has it gotten worse or better?
I don't have data, but it is clearly less than in other ex-Soviet countries. State-run enterprises provide some additional level of job security.