What is known about the distribution of eigenvectors of random matrices?
If you choose the matrix elements of $A$ independently from a Gaussian distribution you have the socalled Ginibre ensemble of random-matrix theory. The statistics of the eigenvalues is known, see for example Eigenvalue statistics of the real Ginibre ensemble. The statistics of the eigenvectors, and the eigenvector-eigenvalue correlations, have been much less studied, I know of just a few papers:
Eigenvector statistics in non-Hermitian random matrix ensembles
Statistical properties of eigenvectors in non-Hermitian Gaussian random matrix ensembles
Correlations of eigenvectors for non-Hermitian random-matrix models
While in the Hermitian ensembles (GOE, GUE) the eigenvectors of different eigenvalues are independent, in the non-Hermitian ensemble eigenvectors are highly correlated if the two eigenvalues lie close in the complex plane.
This is just a small update (much later!) on these interesting questions. On the 4th point of the OP, the distribution of the number of real eigenvalues can be found in my pre-print https://arxiv.org/abs/1512.01449.
Another update: in the paper https://arxiv.org/abs/1608.04923 it is shown how one can use methods from free probability theory to go also beyond the Ginibre ensemble and treat eigenvector correlations for random matrix models corresponding to the single ring theorem