  GR and other metric theories of gravity are formulated with an arbitrary
auxiliary curved background in a Lagrangian formalism. A new sketch of how to
include spinor fields is included. Conserved quantities are obtained using
Noether's theorem and expressed as divergences of antisymmetric densities,
connecting local perturbations with quasi-local conserved quantities. The
background's arbitrariness matches the so-called non-localizability of
gravitational energy (infinity of localizations). The formalism has two partly
overlapping uses: practical applications of pure GR (with fictitious
background) and foundational considerations in which background causality
facilitates quantization. The Schwarzschild solution is a primary application.
Various possibilities for calculating the mass using surface integration are
given. A field-theoretic curved spacetime is given from spatial infinity to the
horizon and even to the true singularity. Trajectories of test particles in the
Schwarzschild geometry are gauge-dependent in that even breakdowns at the
horizon can be suppressed (or generated) by naive gauge transformations. This
fact illustrates the auxiliary nature of the background metric and the need for
some notion of maximal extension---much as with coordinate transformations in
geometric GR. A continuous collapse to a point mass in the field-theoretic
framework is given. The field-theoretic method is generalized to arbitrary
metric theories in $D$ dimensions. The results are developed in the framework
of Lovelock gravity and applied to calculate masses of Schwarzschild-like black
holes. The bimetric formalism makes it natural to consider a graviton mass.
Babak and Grishchuk's numerical and hence nonperturbative work sheds light on
questions of a (dis)continuous massless limit for massive pure spin-2 and the
classical (in)stability of spin-2/spin-0 theory.
