The first two are definitely correct and natural. I am less sure about the third. It could be correct, but I think it would be more usual to say 'Celtic won six straight games', using 'straight' as an adjective. Re the issue of singular and plural: 'Celtic' is the name of a top Scottish football club, pronounced (unusually) with an 's' sound at the beginning. In British English we use plural agreements for sports teams, so the grammar of this sentence is fine. If it's a British context, referring to the Scottish football club, the combination of the singular team name and plural verb is correct; if this refers to the American basketball team known as The (Boston) Celtics, you'd need to add an article and make the team name plural.

They don’t mean the same thing. The following have nearly the same meaning, describing the present Celtic are on a six-game winning streak. Celtic have won six games in a row. Celtic have won six straight games. In the past, The Boston Celtics had a six-game winning streak. The Boston Celtics won six games in a row. The Boston Celtics won six straight games. In the US, “The Celtics” refer to a professional basketball team in Boston, and “Celtic”, the soccer club, wouldn’t be widely known. The reverse is probably true elsewhere.

My apologies, DavidK. I had presumed from the singular name 'Celtic' that this was a British context. I have deleted my comment and amended my answer accordingly.