  A new paradigm for rapid stellarator configuration design has been recently
demonstrated, in which the shapes of quasisymmetric or omnigenous flux surfaces
are computed directly using an expansion in small distance from the magnetic
axis. To further develop this approach, here we derive several other quantities
of interest that can be rapidly computed from this near-axis expansion. First,
the $\nabla\vec{B}$ and $\nabla\nabla\vec{B}$ tensors are computed, which can
be used for direct derivative-based optimization of electromagnetic coil shapes
to achieve the desired magnetic configuration. Moreover, if the norm of these
tensors is large compared to the field strength for a given magnetic field, the
field must have a short length scale, suggesting it may be hard to produce with
coils that are suitably far away. Second, we evaluate the minor radius at which
the flux surface shapes would become singular, providing a lower bound on the
achievable aspect ratio. This bound is also shown to be related to an
equilibrium beta limit. Finally, for configurations that are constructed to
achieve a desired magnetic field strength to first order in the expansion, we
compute the error field that arises due to second order terms.
