ag.algebraic geometry - Questions Suggested by the Parabolic Subgroup Definition

i)-ii) If the subgroup isn't closed, there's no reason in general for the quotient space to even be an algebraic variety. So for the definition to even make sense, there has to be some guarantee that $G/P$ is a variety. A standard result says that if $H subset G$ is a closed subgroup of a linear algebraic group, then the quotient $G/H$ is a quasi-projective variety. You can find these results, for example, in Borel's text on linear algebraic groups. I don't know of an example of a non-closed subgroups whose quotient both exists and is not complete.

(iii) Pretty much by definition, $G/P$ is what's known as a partial flag variety. For the classical groups, you can directly verify that you get flag-like objects in this way, and for more general groups this is taken as the definition. However, if by flag variety you mean the full flag variety, then just pick any parabolic subgroup which is not a Borel subgroup. The simplest example occurs for $G = GL_3$ where you have parabolic subgroups corresponding the variety of lines in $mathbb{C}^3$ (i.e., $mathbb{P}^2$) and its dual, the variety of planes in $mathbb{C}^3.$

(iv) I'm not sure exactly what you're asking for here. Just examples of varieties that aren't quotients of linear algebraic groups?