  The high-precision photometry from the CoRoT and Kepler satellites has led to
measurements of surface rotation periods for tens of thousands of stars. Our
main goal is to derive ages of thousands of field stars using consistent
rotation period measurements in different gyrochronology relations. Multiple
rotation periods are interpreted as surface differential rotation (DR). We
re-analyze the sample of 24,124 Kepler stars from Reinhold et al. (2013) using
different approaches based on the Lomb-Scargle periodogram. Each quarter
(Q1-Q14) is treated individually using a prewhitening approach. Additionally,
the full time series, and different segments thereof are analyzed. For more
than 18,500 stars our results are consistent with the rotation periods from
McQuillan et al. (2014). Thereof, more than 12,300 stars show multiple
significant peaks, which we interpret as DR. Gyrochronology ages between 100
Myr and 10 Gyr were derived for more than 17,000 stars using different
gyrochronology relations. We find a bimodal age distribution for Teff between
3200-4700 K. The derived ages reveal an empirical activity-age relation using
photometric variability as stellar activity proxy. Additionally, we found 1079
stars with extremely stable (mostly short) periods. Half of these periods may
be associated with rotation stabilized by non-eclipsing companions, the other
half might be due to pulsations. The derived gyrochronology ages are well
constrained since more than 93.0 % of the stars seem to be younger than the Sun
where calibration is most reliable. Explaining the bimodality in the age
distribution is challenging, and limits accurate stellar age predictions. The
existence of cool stars with almost constant rotation period over more than
three years of observation might be explained by synchronization with stellar
companions, or a dynamo mechanism keeping the spot configurations extremely
stable.
