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. 

However, you cannot expect that they are locally complete intersections (l.c.i.) in general. For instance, the weighted projective plane $\mathbb{P}(1, \, 1, \, 2)$ is isomorphic to a quadric cone, so it is a l.c.i. projective variety, whereas $\mathbb{P}(1, \, 1, \,3)$ is isomorphic to a cone over a twisted cubic, hence it has a $\frac{1}{3}(1, \, 1)$-singularity at its vertex, which is not a complete intersection singularity.

Furthermore, 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.