I am new to SharePoint. My boss wants me to make a certain template for our client (which will be completely custom made), and he also wants to use the Sharepoint webapps and webparts after the branding. However from what I know by my limited knowledge is that branding can be done only in Publishing website and publishing websites does not support OOTB SharePoint webapps and features. But i would like to confirm from more experienced people if it really is so, before telling my boss its impossible. Thank You.
No, the branding can be done on the websites not only publishing site.
The publishing site supports the OOTB SharePoint webapps and features.
It is possible to brand a SharePoint online site and also use SharePoint inbuilt apps like Hero, weather forecast and so on.
The SharePoint inbuilt apps like Hero and weather forecast can be added in the SharePoint modern page. They have nothing to do with the publishing feature.
You could create the site page in the site pages library, then edit the site page and add the Hero web part and weather forecast web part in the site page.