Regarding your question about weighted projective spaces, a lot is known about them, see for instance **[1]** and **[2]**.

In particular, any weighted projective space $\mathbb{P}(Q)$ is irreducible, normal, Cohen-Macauley and has at most cyclic quotient singularities (hence rational singularities), see **[2]**, p. 122. 

Moreover, any smooth $\mathbb{P}(Q)$ is isomorphic to $\mathbb{P}^r$, see **[1]**, p. 39

**References.**

**[1]** I. Dolgachev: *Weighted projective varieties*, Group actions and vector fields, Lecture Notes in Math. **956** (1982), 34-71. 

**[2]** M. Beltrametti, L. Robbiano: *Introduction to the theory of weighted projective spaces*, Expo. Math. **4** (1986), 111-162.