Take the following definition:
"A *parabolic subgroup* of a linear algebraic group defined over $\mathbb{C}$ is a subgroup, closed in the Zariski topology, for which the quotient space is a projective algebraic variety."

My questions are:

(i) Why include *closed* in the definition?

(ii) What is an example of a projective algebraic variety that is the quotient of a linear algebraic group by a non-Zariski-closed subgroup?

(iii) What is an example of a quotient by a parabolic subgroup that is not a flag manifold?

(iv) Elliptic curves cannot be described as quotients of linear algebraic groups. What are other examples of families of varieties that cannot be expressed in this form?