The Arduino does not have built in clock but there are various Real Time Clocks that you can add very cheaply. There is no reason the Arduino cannot turn the power on and off to the televisions using the relay module you suggested, on a time basis, and it could keep a log on an SD card.
However rather than pulling back upto sixteen power cables to the relay module it might be worth considering having a relay at each television with one communications cable running around them or perhaps a wireless connection. I have not searched for them but there are bound to be products around that you could drive from an Arduino. You should work out roughly the length of cable run required.
You have also muddied the water a bit by mentioning Windows. The Arduino does not have any operating system and certainly not Windows. If you want/need Windows then you might be better controlling the TVs from the Windows machine.