Does Every Country Have a Bishop?

Basically, yes, every country falls under the authority of a bishop. However that is not exactly the same as every country having a functioning bishop. For example, Saudi Arabia has no Catholic churches whatsoever but the territory of the country technically falls under the authority of Apostolic Vicariate of Northern Arabia which is based in Bahrain. It is not a diocese but rather is a territory that falls under the authority of an office of the Church which is headed by a Bishop.

Every diocese has a Bishop. If a country has no dioceses (like Saudi Arabia) it is usually deemed mission territory and is placed under the authority of a religious order or Church Congregation.