Re: How launch actions/handler from share (MenuItem or widget)?
I did not say to literally use "alfresco/menus/AlfMenuItemMixin". The "alfresco/header/AlfMenuItem" you used initially is an extension of "alfresco/menus/AlfMenuItemMixin" and as such supports publishing events. So, simply use "alfresco/header/AlfMenuItem" and configure that with the publish topic etc.
Note that "propertyToRender" is inapplicable to any element in a menu. You also cannot expect "processCurrentItemTokens" to work, because the header menu does not have a "currentItem" state variable.
I can only advise to work through the Aikau tutorial and to use the JSDoc of the modules to better understand them.