Is it possible in any way to set my computer to turn on automatically at the beginning of a recording and turn off automatically at the end of the recording so my computer doesn’t have to be left on all the time?
If your computer is configured to automatically log in, then switching on could be as simple as a wall plug timer.
For switching off, see the “power management” features of you operating system. Most operating systems have settings that allow the computer to shut down after being idle for a specified period.
For the actual recording it would be easier to run a command line program (such as SoX http://sox.sourceforge.net/) and set it to start as a “scheduled task” rather than trying to operate a graphical application such as Audacity.
Alternatively you could miss out the computer altogether and rig up an old VHS video player to record audio.