Are there freeware programs that will automatically start up and shut down my PC at various times of the day?

Answer 1

There are multiple programs found in the internet that could schedule the shutdown of your PC.

However, starting up your computer on a scheduled program inside your your computer would be impossible, as a shut down PC is well, shut down.

But, there are programs that could "Stand By/Hibernate" your computer and then resume normal operation on specific times of your choosing.

Another thing that you can do is you can start up your computer by using wake up (Wake On LAN) commands from another PC, however this requires a network card that supports the feature. also, those programs do not seem to work over the internet.