Can slick automatically create tables in the database (generate SQL and execute) from the models?

You can create tables in Slick from a model: it's not related to the codegen tool, though.

When you define a model in Slick, you can use the .schema method to generate the database schema commands. There are examples of this in The Slick Manual:

 // Assuming we have coffees and suppliers queries, we combine the schemas:
 val schema = coffees.schema ++ suppliers.schema


 // Now we can run a variety of commands to CREATE TABLE etc:
 db.run(DBIO.seq(
   schema.create,
   schema.createIfNotExists,
   schema.drop,
   schema.dropIfExists
))

However, that's not automatic: you'd need to write something in your start-up code to decide to run the DDL commands or not.