I'm turning here (a variation of) a question asked by a friend of mine. For the purposes of this question I will say that a compact complex manifold is projective if it is isomorphic to a subvariety of $\mathbb{P}^n$ and algebraic if it is isomorphic to the complex analytic space associated to a scheme.

One often finds many examples of compact complex manifolds which are not projective. For instance Hopf and Inoue surfaces, some K3, some tori... Usually the proof shows that either this manifold is not Kahler, or the Kahler cone does not intersect the lattice of integral cohomology. But of course this does not tell us anything about their being algebraic in the sense outlined above.

So there are two questions. First, what is an example of a compact complex manifold which is not algebraic? Probably this is standard, but I don't have any reference in mind. I think that the complex analytic space associated to a smooth algebraic space which is not a scheme will do, but I'm not expert of algebraic spaces, so I don't have even such an example in mind.

The second, subtler, question is: are the examples above of nonprojective complex manifold also non algebraic? How one can prove such a statement?