I answer only your last question.

Take a finite-dimensional $\mathbb C$-linear space $V$ equipped with a positive-definite hermitian form $\langle-,-\rangle$. The tangent space $\text{T}_p{\mathbb P}_{\mathbb C}V$ at a $1$-dimensional linear subspace $p\subset V$ is naturally identified with $\text{Lin}_{\mathbb C}(p,V/p)$ which in turn is naturally identified with $\text{Lin}_{\mathbb C}(p,p^\perp)$ with respect to the orthogonal decomposition $V=p\oplus p^\perp$. Using this decomposition, we can assume that
$\text{Lin}_{\mathbb C}(p,p^\perp)\subset\text{Lin}_{\mathbb C}(V,V)$ (extending by zeros). The hermitian form on $\text{Lin}_{\mathbb C}(V,V)$ given by the rule $\langle t_1,t_2\rangle=\text{tr}(t_1\circ t_2^*)$ induces on
${\mathbb P}_{\mathbb C}V$ the famous Fubibi-Study hermitian structure. Naturally, the distance can be expressed in terms of the original hermitian form on $V$. The distance in the projective space between the points
$0\ne p_1,p_2\in V$ is given by
$\text{dist}(p_1,p_2)=\text{arccos}\frac{\langle p_1,p_2\rangle\langle p_2,p_1\rangle}{\langle p_1,p_1\rangle\langle p_2,p_2\rangle}$.

A similar approach works for many hyperbolic geometries (including the complex hyperbolic one), elliptic geometries, and many others that come from grassmannians.