Yes. Democracy (government by the people) was the ancient Athenian ideal of direct government.
Republic (res publica, administration of public "things"/affairs) is the roman inspired version - but it just means that the country is considered to be governed as a whole "public thing" by some body, as opposed to being a thing belonging to the emperor or king, etc. to do as they wish and for private gate).
A republic doesn't have to be democratic - a cadre of "wise people" could e.g. govern one to the exclusion of the public at large. Or the public might be given some token participation (like voting for one of several bodies of government, or merely voting for electors to elect the actual government, etc., and/or, common too, voting for people with fixed 4-5 year terms on abstract platforms for the whole term taken wholesale).
The discussion is blurred because modern republics (almost everyone) often label themselves democracies because "vote".
No, that's how a republic might work. In democracy companies are not allowed to yield such power, and people elected can be revoked at any time....