CARMENES input catalog of M dwarfs: VII. New rotation periods for the survey stars and their correlations with stellar activity

Abridged: We measured photometric and spectroscopic P_ rot for a large sample of nearby bright M dwarfs with spectral types from M0 to M9, as part of our continual effort to fully characterize the Guaranteed Time Observation programme stars of the CARMENES survey. We determine P_ rot for 129 stars. Combined with the literature, we tabulate P_ rot for 261 stars, or 75 this sample by comparing them with activity signatures and checking for consistency between multiple measurements. We find that 166 of these stars have independent evidence that confirmed their P_ rot. There are inconsistencies in 27 periods, which we classify as debated. A further 68 periods are identified as provisional detections that could benefit from independent verification. We provide an empirical relation for the P_ rot uncertainty as a function of the P_ rot value, based on the dispersion of the measurements. We show that published formal errors seem to be often underestimated for periods ≳ 10 d. We highlight the importance of independent verification on P_ rot measurements, especially for inactive M dwarfs. We examine rotation-activity relations with emission in X-rays, Hα, Ca II H K, and surface magnetic field strengths. We find overall agreement with previous works, as well as tentative differences in the partially versus fully convective subsamples. We show P_ rot as a function of stellar mass, age, and galactic kinematics. With the notable exception of three transiting planet systems and TZ Ari, all known planet hosts in this sample have P_ rot≳ 15 d. This indicates that important limitations need to be overcome before the radial velocity technique can be routinely used to detect and study planets around young and active stars.