Is the quotient of a smooth complex projective surface by an involution projective? Suppose the quotient happens to be smooth; does that change the situation?
The quotient of a quasi-projective variety $X$ by a finite group action $G$ is quasi-projective. As is explained in [1, Remarque V.5.1], this follows from [1, Theoreme V.4.1].
 M. Demazure and A. Grothendieck. Schémas en groupes I, II, III (SGA 3). Lecture Notes in Math. 151, 152, 153. Springer-Verlag, New York, 1970