Move data from MySQL 5.0 to other server with MySQL 5.7 We are switching servers from a Windows Server 2003 (32-bit) with a MySQL 5.0.51b to a Windows Server 2016 (64-bit) on which I will install the latest MySQL version (5.7).

The database contains just tables (no views, foreign keys...) which I want to move to the new database. Should I be able to use mysqldump and mysqlimport (as per the documentation) even though the difference in MySQL versions?
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.