Hmmm think I have made some progress here...

I went back to the copy of the domain\user3 account and logged into the named instance. I then went to create a new table in the designer, and as it did before, it used the dbo schema. OK, so then I issued the following T-SQL statement:

CREATE TABLE TestTable (ID INT NULL)

This then created a table with the schema domain\user3copy, and also created the database user and schema with that name. Then I went to create another table with the designer and this time it used the domain\user3copy schema.

If I now delete the new table and domain\user3copy database user and schema that was just created and then create a new table in the designer it uses the dbo schema. However, if I try to change the table in the designer (once I have saved it) it won't let me and tells me the table is read only.

So does this mean I have to make the Windows Group a member of db_owner? I don't want to do this because it would give the users more privileges than I want to allow. Such as drop database!