SSMS: possible to script 'file per object' into categorised folders like in object explorer?
You could import the database definition in SSDT using the "Schema Compare" tool. 

Simply create an empty database project, right click and select "Schema compare". Select the project as target and the live database as source. Click "Compare" and then "Update": SSDT will script all objects and add them to your project.

SSDT by default creates a folder for each schema and in each folder creates separate folders for tables, stored procedures, views and so on.