Use-case of default Teamwebsite? Can we use it, or should we create a new one?

You could just use the root site for all your documents but you should consider security, discoverability, offline syncing and other use cases for O365 services. I would create separate sites for at least each department as a simple information architecture so you can set permissions to all content in that site and not on a folder or file level as you will hit a unique permission limit, consider for instance the whole organisation probably shouldnt see HR or finance documents.
You will probably find users will get more productivity gains from using other tools such as Planner for tasks and Teams for conversations that are associated with O365 groups. I regularly see users manage tasks in spreadsheets where Planner would be better and email distribution lists that would be better in Teams.
Think longer term on how the services might be used even if your use case at the moment is just files. Teams will also help with keeping different sites easily accessible in one place.
Also many people use the home site as an intranet home page with news, events and other corporate information that is managed by a smaller team so in this case you wouldn't want all the files and permissions for everyone to edit.