Just to be clear, you can download and burn whatever whenever wherever. But, for Avast Free (or paid editions too), you don’t really need to (except, maybe, your license when you use the paid editions).

Do your job/tasks, and when you are finished, download the latest version available and install it.

About Avast automatically updating itself, what you have been seeing is the engine and antivirus database being updated. The program is updated once in a while, and it should show you a tray notification about it, so to ask you if you want to update to the latest program version. At least, this is the default setting.

Either you (dis)missed those tray notifications, or for some reason the tray notifications were delayed (not displayed at all), which is still a possibility already seen during this year.

HTH to clarify.