  The occurrence of high temperature superconductivity, and the competition
with magnetism, in stoichiometric and doped LaOFeAs and isostructural
iron-oxypnictides is raising many fundamental questions about the electronic
structure and magnetic interactions in this class of materials. There are now
sufficient experimental data that it may be possible to identify the important
issues whose resolution will lead to the understanding of this system. In this
paper we address a number of the important issues. One important characteristic
is the Fe-As distance (or more abstractly the pnictogen (Pn) height $z$(Pn));
we present results for the effect of $z$(Pn) on the electronic structure,
energetics, and Fe magnetic moment. We also study LaOFeAs under pressure, and
investigate the effects of both electron and hole doping within the virtual
crystal approximation. The electric field gradients for all atoms in the
LaOFeAs compound are presented (undoped and doped) and compared with available
data. The observed $(\pi,\pi,\pi)$ magnetic order is studied and compared with
the computationally simpler $(\pi,\pi,0)$ order which is probably a very good
model in most respects. We investigate the crucial role of the pnictogen atom
in this class, and predict the structures and properties of the N and Sb
counterparts that have not yet been reported experimentally. At a certain
volume a gap opens at the Fermi level in LaOFeN, separating bonding from
antibonding bands and suggesting directions for a better simple understanding
of the seemingly intricate electronic structure of this system. Finally, we
address briefly on the possible effects of post-lanthanum rare earths, which
have been observed to enhance the superconducting critical temperature
substantially.
