I bought a flat through a real estate company (SCI). Does the 10-day cooling-off period apply?

It depends. The 10-day withdrawal right applies to non-professional buyers (Article L 271-1 CCH). If the SCI’s purpose includes real estate acquisition, it may be deemed professional (Cass. 3rd Civ., 24 Oct. 2012, No. 11-18.774).