Yes that is a good idea, i have it...
Software: the simplest option is to install WAMP (Windows), LAMP (Linux), or MAMP (Mac), which includes Apache, MySQL and PHP. Or you can do whatever you want with customizing that or just installing a basic server (probably Apache).
Yes,, running this system around the clock will cause any system crashes ? cooling is sufficient ? which is the best temp to keep the server.
Of course you'll need to learn how to run/manage a server as well, but you can figure that out as you go. Note that you'll need to run the software 24/7 if you want your website to be available all the time.