The Fermi Large Area Telescope (LAT) has provided the most detailed view to date of the emission toward the Galactic center (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 1-100GeV from a 15{deg}x15{deg} region about the direction of the GC. Specialized interstellar emission models (IEMs) are constructed to enable the separation of the {gamma}-ray emissions produced by cosmic ray particles interacting with the interstellar gas and radiation fields in the Milky Way into that from the inner ~1kpc surrounding the GC, and that from the rest of the Galaxy. A catalog of point sources for the 15{deg}x15{deg} region is self-consistently constructed using these IEMs: the First Fermi-LAT Inner Galaxy Point Source Catalog (1FIG). The spatial locations, fluxes, and spectral properties of the 1FIG sources are presented, and compared with {gamma}-ray point sources over the same region taken from existing catalogs. After subtracting the interstellar emission and point-source contributions a residual is found. If templates that peak toward the GC are used to model the positive residual the agreement with the data improves, but none of the additional templates tried account for all of its spatial structure. The spectrum of the positive residual modeled with these templates has a strong dependence on the choice of IEM.