Regarding Question 1: Of course, this generalizes to the space $\mathrm{S}_{pq}(V)$ of all splittings of a vector space $V$ into subspaces $V = P\oplus Q$ where $\dim P = p>0$ and $\dim Q = q>0$ and, of course, $\dim V = p+q$.

The tangent space at $(P,Q)\in \mathrm{S}_{pq}(V)$ is canonically isomorphic to $(P{\otimes}Q^*)\oplus (Q\otimes P^*)$, for the same reasons that you list in the case where $p$ or $q$ is $1$, so the same reasoning applies. When the ground field is $\mathbb{R}$, this is an irreducible pseudo-Riemannian symmetric space (of split type), and, as such appears in Berger's classification of pseudo-Riemannian symmetric spaces as $\mathrm{SL}(V)/\mathrm{S}(\mathrm{GL}(P){\times}\mathrm{GL}(Q))$.

When the ground field is $\mathbb{C}$, this space can be regarded naturally as a complexification of the space $\mathrm{Gr}_p(V)$ of $p$-planes in $V$. This canonical metric then turns out to be, in an appropriate sense, the holomorphic extension of the Fubini-Study metric on $\mathrm{Gr}_p(V)$ (defined relative to any positive definite Hermitian inner product on $V$) to this complexification.

Regarding Question 2: The metric (as a quadratic form) does not extend continuously to the product $\mathrm{Gr}_p(V)\times\mathrm{Gr}_q(V)$. The reason is that the volume form of the canonical metric on $\mathrm{S}_{pq}(V)$ gives $\mathrm{S}_{pq}(V)$ infinite volume, which it could not do if the quadratic form extended continuously across the incidence hypersurface. (Just look at the case $p=q=1$ to convince yourself of this.)

Regarding Question 1: Of course, this generalizes to the space $\mathrm{S}_{pq}(V)$ of all splittings of a vector space $V$ into subspaces $V = P\oplus Q$ where $\dim P = p>0$ and $\dim Q = q>0$ and, of course, $\dim V = p+q$.

The tangent space at $(P,Q)\in \mathrm{S}_{pq}(V)$ is canonically isomorphic to $(P{\otimes}Q^*)\oplus (Q{\otimes}P^*)$, for the same reasons that you list in the case where $p$ or $q$ is $1$, so the same reasoning applies.

When the ground field is $\mathbb{R}$, this is an irreducible pseudo-Riemannian symmetric space (of split type), and, as such, appears in Berger's classification of pseudo-Riemannian symmetric spaces as $\mathrm{SL}(V)/\mathrm{S}(\mathrm{GL}(P){\times}\mathrm{GL}(Q))$.

When the ground field is $\mathbb{C}$, this space can be regarded naturally as a complexification of the space $\mathrm{Gr}_p(V)$ of $p$-planes in $V$. This canonical metric then turns out to be, in an appropriate sense, the holomorphic extension of the Fubini-Study metric on $\mathrm{Gr}_p(V)$ (defined relative to any positive definite Hermitian inner product on $V$) to this complexification.

Regarding Question 2: The metric (as a quadratic form) does not extend continuously to the product $\mathrm{Gr}_p(V)\times\mathrm{Gr}_q(V)$. The reason is that the volume form of the canonical metric on $\mathrm{S}_{pq}(V)$ gives $\mathrm{S}_{pq}(V)$ infinite volume, which it could not do if the quadratic form extended continuously across the incidence hypersurface. (Just look at the case $p=q=1$ to convince yourself of this.)