The answer to this question depends on how it is read. As there is no regime which only has aspects of Communism in it and no other political and/or economic philosophy, a "purely communist" country is a hypothetical country. It could refer to a country fully in-line with the Communist Manifesto OR it could refer to the best possible implementation of Communism based on the historic regimes associated with Communism. These two hypothetical countries would be radically different.
Communist Manifesto Country: The Communist Manifesto describes a transformation of the state through several iterations. The state would first start as a capitalist and industrialist state. Then the workers of that state would rise up in a revolution and become a dictatorial legislature called the Politburo. In this second stage, the Dictatorship of the Proletariat, as it is called, would invest massively in internal education, Propaganda, social services, and wealth redistribution. It would plan the economy and improve the rights of the disenfranchised poor and weaken the privileges of the wealthy. Eventually, this should result in creating a highly moral but economically average population. At that time, the Dictatorship would disband according to anarcho-syndicalist principles and the people would simply self-govern according to the high level of morality that they had attained.
Communism In Reality: Cuba is probably the purest example of Communism in the world. It has a dictatorial leadership with minimal citizen participation, a planned economy (and an illegal economy that accounts for needs not met by the planned economy), a strong provision of social services, education, and welfare, low wages, and low production. Since Cuba was agrarian pre-revolution, it has not industrialized because capitalists are deterred from investing. Communists will generally argue that Cuba is in the midst of the Dictatorship of the Proletariat stage, even though Cuba does not have a Politburo and there is no indication that the Cuban dictatorship is willing to surrender power should the population be "moral enough" to self-govern.
In general, no. The position of General Secretary is more analogous to Prime Minister than President. Any position labeled "President" in a communist country is generally a purely ceremonial position.
No, it isn't a communist country.
No, Chad is not a communist country.
London is not a country. It is a city. It is in England, which is not a communist country.
No, Iceland is not a communist country.
Yugoslavia is no longer a country. Before its breakup it was a Communist country.
China is a communist country.
Thailand is not a Communist Country
No, it isn't a communist country.
No, Brunei is not a communist country.
Cuba is the only communist country in the western Hemisphere.
North is not a country. If you are referring to North Korea, then yes, North Korea is a communist country.