Avast Mac Security installs its real time protection components as kext (we call them “shield” in AV UI). They run regardless of the icon in the dock/menu bar. (Of course, the shield shouldn’t be “disabled” in Preferences ;))
As Pondus and Eddy said - “install and forget”. The product updates automatically and we let you know when there is a malicious website, download or program trying to get into your Mac.