  The Fermi Large Area Telescope (LAT) has provided the most detailed view to
date of the emission towards the Galactic centre (GC) in high-energy
gamma-rays. This paper describes the analysis of data taken during the first 62
months of the mission in the energy range 1-100 GeV from a $15^\circ \times
15^\circ$ region about the direction of the GC, and implications for the
interstellar emissions produced by cosmic ray (CR) particles interacting with
the gas and radiation fields in the inner Galaxy and for the point sources
detected. Specialised interstellar emission models (IEMs) are constructed that
enable separation of the gamma-ray emission from the inner $\sim 1$ kpc about
the GC from the fore- and background emission from the Galaxy. Based on these
models, the interstellar emission from CR electrons interacting with the
interstellar radiation field via the inverse Compton (IC) process and CR nuclei
inelastically scattering off the gas producing gamma-rays via $\pi^0$ decays
from the inner $\sim 1$ kpc is determined. The IC contribution is found to be
dominant in the region and strongly enhanced compared to previous studies. A
catalog of point sources for the $15^\circ \times 15^\circ$ region is
self-consistently constructed using these IEMs: the First Fermi-LAT Inner
Galaxy point source Catalog (1FIG). After subtracting the interstellar emission
and point-source contributions from the data a residual is found that is a
sub-dominant fraction of the total flux. If spatial templates that peak toward
the GC are used to model the positive residual and included in the total model
for the $15^\circ \times 15^\circ$ region, the agreement with the data
improves, but none of the additional templates account for all of the residual
structure. The spectrum of the positive residual modelled with these templates
has a strong dependence on the choice of IEM. [Abridged]
