metric and measure on the projective space

Apparently you ask about metric as two-point function, not about Riemannian metric in the projective space. The answer (consistent with the Riemannian metric) is: angle between lines through the origin in $\mathbb R^{n+1}$ (which represent points in $\mathbb RP^n$). The diameter of the space is $\pi/2$.

Correct on both counts. Using the Euclidean metric on $\mathbb R^{n+1}\setminus \{0\}$ and collapsing along lines leads to a disaster: the quotient pseudo-metric is identically zero, because the Euclidean distance between any two lines through the origin is zero. This issue does not arise with $S^n/\{\pm I\}$, whether we use the restriction (chord) metric on $S^n$ or the intrinsic (arc) metric. The latter choice is more natural, and yields the metric described in 1.

Pushing things from $S^n$ to $S^n/\{\pm I\}$ by the quotient map is a convenient and natural way to equip $\mathbb RP^n$ with various structures, including the measure. The pushforward of normalized Lebesgue measure on $S^n$ gives the standard probability measure on $\mathbb RP^n$.

First, try pushing forward by the quotient map. Sometimes this does not work: for example, pushing the Lebesgue measure by the quotient map $\mathbb R\mapsto \mathbb R/\mathbb Z$ we get something ugly (which gives infinite measure to every nontrivial arc of the circle). In such a case, consider restricting the measure to a fundamental domain first.