Can you own your own business in a communist country?
In a truly communist country there would be no "businesses".
The world has never seen a truly communist country of any significance, though, and in most countries considered "communist" you can, kinda sorta, own your own business. Though if it does very well the government may step in.
It's different from country to country. In a communist country, it's not normally accepted to want to choose your own religion. In a capitalist country it is encouraged to choose one's own religion. As a human being I would say yes, it is important to choose your own religion and strongly follow your own beliefs.
NO. Turkey was founded as a right-wing quasi-fascist republic. After World War II, communist guerillas, tried to take over the Turkish Government, but they were prevented by the US-backed Turkish military. Turkey was considered a strategic US ally in the Cold War and was never allowed to fall to Communism.
There is no single answer to that. There are a number of different communist countries and there are different kinds of doctors, so there would be a broad range of pay rates. They also have their own currencies and economies and other aspects of society are different, like the services available and the cost of living, so you can't compare them to you own country or even to each other.