Is a sovereign state always a country?

For the United States -- Yes. Sovereign states ARE countries. An example to prove this is Puerto Rico. They are a US Territory. There was a vote several years ago to declare them a sovereign state - which would have made their own country. It didn't pass and they are still a territory and are known as the Commonwealth of Puerto Rico.

However, it depends on who you talk to in the rest of the world. For instance, Pakistan does not recognize Armenia as a sovereign state. Now think of Israel and Palestine. Neither recognize each other as sovereign nations.

Then there's the UN, which is recognized as the central governing body for the world. There's all sorts of sovereignty issues there.