The complex moduli space does not admit a toric strucutre, since the orbifold fundamental group of a toric orbifold must be abelian. Indeed, $\pi_1(\mathbb C^*)^n$ surjects on the orbifold fundamental group. Also, the orbifold stabisier of each point on a toric orbifold is a finite abelian group. At the same time the stabiliser of the quintic $\sum_i z^5=0$ is a non-comutative group. Also I am sure that the orbifold fundamental group of the moduli space of quintics contains free (non-abelian) subgroups, but I don't know how to prove it.

Also it should be true that the Tiechmuller space is not algebraic. It least this happen in lower dimensions for cubics in $\mathbb CP^2$ and for quartics in $\mathbb CP^3$. In the first case the Theichmuiller space is a disk, and in the second it is a domain are hermitian domains of type IV. Moduli spaces of polarised K3 are discussed here for example, here:

http://people.bath.ac.uk/masgks/Papers/k3moduli.pdf

The complex moduli space does not admit a toric strucutre, since the orbifold fundamental group of a toric orbifold must be abelian. Indeed, $\pi_1(\mathbb C^*)^n$ surjects on the orbifold fundamental group. Also, the orbifold stabisier of each point on a toric orbifold is a finite abelian group. At the same time the stabiliser of the quintic $\sum_i z^5=0$ is a non-comutative group. Also I am sure that the orbifold fundamental group of the moduli space of quintics contains free (non-abelian) subgroups, but I don't know how to prove it.

Also it should be true that the Tiechmuller space is not algebraic. It least this happen in lower dimensions for cubics in $\mathbb CP^2$ and for quartics in $\mathbb CP^3$. In the first case the Theichmuiller space is a disk, and in the second it is 
a hermitian domain of type IV. Moduli spaces of polarised K3 are discussed here for example, here:

http://people.bath.ac.uk/masgks/Papers/k3moduli.pdf