We present a study of NGC 4993, the host galaxy of the GW170817 gravitational-wave event, the GRB 170817A short gamma-ray burst (sGRB), and the AT 2017gfo kilonova. We use Dark Energy Camera imaging, AAT spectra, and publicly available data, relating our findings to binary neutron star (BNS) formation scenarios and merger delay timescales. NGC 4993 is a nearby early-type galaxy, with an i-band Sérsic index n = 4.0 and low asymmetry (A = 0.04 ± 0.01). These properties are unusual for sGRB hosts. However, NGC 4993 presents shell-like structures and dust lanes indicative of a recent galaxy merger, with the optical transient located close to a shell. We constrain the star formation history (SFH) of the galaxy assuming that the galaxy merger produced a star formation burst, but find little to no ongoing star formation in either spatially resolved broadband SED or spectral fitting. We use the best-fit SFH to estimate the BNS merger rate in this type of galaxy, as R NSM gal = 5.7 -3.3 +0.57 times 10-6 yr-1. If star formation is the only considered BNS formation scenario, the expected number of BNS mergers from early-type galaxies detectable with LIGO during its first two observing seasons is 0.038 -0.022 +0.004, as opposed to ~0.5 from all galaxy types. Hypothesizing that the binary formed due to dynamical interactions during the galaxy merger, the subsequent time elapsed can constrain the delay time of the BNS coalescence. By using velocity dispersion estimates and the position of the shells, we find that the galaxy merger occurred t mer ≲ 200 Myr prior to the BNS coalescence.