1) You're right, it is solved exactly for single electron atoms.

As far as the validity of the approximate solutions, I've read:

"No exact solution of the Schroedinger equation is possible for any of the atoms heavier than hydrogen, but methods of successive approximations can be used to obtain very good approximate solutions to the Schroedinger equations which describe the electrons in heavier atoms. Modern digital computers are almost mandatory in the very laborious calculations required to obtain accurate results for many-electron atoms by successive approximations."

As I understand it, by approximations they don't mean chemical fudge factors, but numerical iterative methods.

2) As Mitch pointed out, I meant adding protons as you move across a period so Zeff is increasing. Now if you stop on an element like O and just keep adding electrons to get to O-, then Zeff decreases but is still strong enough in this case. If gets to O2- or O3- , like you say Zeff is too diluted and so on.

3) I guess you can put it that way, unstable N- can be used to illustrate the relative stability of half filled shells.

What it boils down to with Hund's rule is that maximum multiplicity, in half filled shells, lies lowest in energy because:

" a symmetric spin state forces an antisymmetric spatial state where the electrons are on average further apart and provide less shielding for each other, yielding a lower energy. "