  The nearby open cluster NGC 752 presents a rare opportunity to study stellar
properties at ages >1 Gyr. However, constructing a membership catalog for it is
challenging; most surveys have been limited to identifying its giants and dwarf
members earlier than mid-K. We supplement past membership catalogs with
candidates selected with updated photometric and proper-motion criteria,
generating a list of 258 members, a >50% increase over previous catalogs. Using
a Bayesian framework to fit MESA Isochrones & Stellar Tracks evolutionary
models to literature photometry and the Tycho-Gaia Astrometric Solution data
available for 59 cluster members, we infer the age of, and distance to, NGC
752: 1.34$\pm$0.06 Gyr and 438$_{-6}^{+8}$ pc. We also report the results of
our optical monitoring of the cluster using the Palomar Transient Factory. We
obtain rotation periods for 12 K and M cluster members, the first periods
measured for such low-mass stars with a well-constrained age >1 Gyr. We compare
these new periods to data from the younger clusters Praesepe and NGC 6811, and
to a theoretical model for angular-momentum loss, to examine stellar spin down
for low-mass stars over their first 1.3 Gyr. While on average NGC 752 stars are
rotating more slowly than their younger counterparts, the difference is not
significant. Finally, we use our spectroscopic observations to measure Halpha
for cluster stars, finding that members earlier than $\approx$M2 are
magnetically inactive, as expected at this age. Forthcoming Gaia data should
solidify and extend the membership of NGC 752 to lower masses, thereby
increasing its importance for studies of low-mass stars.

Solar and Stellar Astrophysics;Astrophysics of Galaxies