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 Fubini-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.
Late extension. It is not necessary to require the form $\langle-,-\rangle$ to be positive-definite, and one can also take $p\subset V$ of a fixed dimension, say, $k$, thus dealing with the grassmannian $\text{Gr}_{\mathbb C}(k,V)$. As above, the loci of non-degenerate points get their pseudo-hermitian structure. Many objects of these geometries (such as geodesics, bisectors, etc) admit an easy linear/hermitian description. The degenerate points form the absolute whose geometry can also be described in the hermitian/linear terms. (It is also possible to take $\mathbb R$ or $\mathbb H$ (or even simetimes $\mathbb O$) in place of $\mathbb C$.)