Does 0patch Console have to be running in order for patching to work?

No, neither 0patch Console (0patchConsole.exe, the user interface app) nor 0patch Tray (0patchTray.exe, the 0patch icon in your tray / notification area) need to be running for patching to work.

The patching is done by 0patch Service and 0patch Driver, which are running in the background all the time, and automatically start every time you start Windows.