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.