Move data from MySQL 5.0 to other server with MySQL 5.7
mysqldump and mysqlimport (as described by the documentation you cited) should be fine for your regular tables, especially since you don't have any foreign keys. 

However, due to the differences in MySQL versions, the tables in the mysql database will be different, so I suspect migrating that database probably can't be done with this particular method.

If you need to migrate the mysql database, there are better ways of doing that.