We present results for the short-term variability of Binary Trans-Neptunian Objects (BTNOs). We performed CCD photometric observations using the 3.58m Telescopio Nazionale Galileo (TNG), the 1.5m Sierra Nevada Observatory (OSN) telescope, and the 1.23m Centro Astronomico Hispano Aleman (CAHA) telescope at Calar Alto Observatory. We present results based on five years of observations and report the short-term variability of six BTNOs. Our sample contains three classical objects: (174567) 2003 MW_12_, or Varda, (120347) 2004 SB_60_, or Salacia, and 2002 VT_130_; one detached disk object: (229762) 2007 UK_126_; and two resonant objects: (341520) 2007 TY_430_ and (38628) 2000 EB_173_, or Huya. For each target, possible rotational periods and/or photometric amplitudes are reported. We also derived some physical properties from their lightcurves, such as density, primary and secondary sizes, and albedo. We compiled and analyzed a vast lightcurve database for Trans-Neptunian Objects (TNOs) including centaurs to determine the lightcurve amplitude and spin frequency distributions for the binary and non-binary populations. The mean rotational periods, from the Maxwellian fits to the frequency distributions, are 8.63+/-0.52h for the entire sample, 8.37+/-0.58h for the sample without the binary population, and 10.11+/-1.19h for the binary population alone. Because the centaurs are collisionally more evolved, their rotational periods might not be so primordial. We computed a mean rotational period, from the Maxwellian fit, of 8.86+/-0.58h for the sample without the centaur population, and of 8.64+/-0.67h considering a sample without the binary and the centaur populations. According to this analysis, regular TNOs spin faster than binaries, which is compatible with the tidal interaction of the binaries. Finally, we examined possible formation models for several systems studied in this work and by our team in previous papers.