Projective equivalence: Linear subspaces under the action of $PGL_n$

Solution 1:

1) As far as the $PGL(n+1)$ action is concerned, the general position is given exactly by what you describe: the spans having maximal dimension.

2) If you want the action to be transitive on a tuple of subspaces of dimension $(d_1, \dots, d_k)$, that's the same as saying a fixed choice of subspaces $\Lambda_i^* \in Gr(d_i+1, n+1)$ can be mapped to an arbitrary tuple $(\Lambda_1, \dots, \Lambda_k)$. By a simple dimension count, this should happen when $\sum \dim Gr(d_i + 1, n+1) \le \dim PGL(n+1)$, or $\sum (d_i+1)(n-d_i) \le n(n+2)$. So for points, each $d_i = 0$, and hence the action is transitive as long as $kn \le n(n+2)$ or $k \le n+2$.