Able to open downloaded programs, but once closed need to download again In particular, I'm trying to use Calibre for eBook management. Interestingly, this has never happened with any other app before, but when I   quit the app (CommandQ), it's almost as if I never installed it in the first place.

Any insight into what is going on?
You might be running them from their DMG file, usually located in the folder where you set your downloads to go to. 

Apps you want in your /Applications folder, you have to copy them there. That's why most DMGs provide Applications' alias in the DMG. 

Use spotlight or  apps like Easyfind to find the relevant installer.