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Welcome to the forum.
Unfortunately, this doesn't appear to be the correct forum for this question. You would probably want to post this either in the APEX forum or the SQL Developer forum. SQL Developer doesn't use ODBC.
That said, I don't believe you'll be able to connect to the database behind apex.oracle.com directly. First off, APEX users and workspaces are (in general) completely separate from database users. You would need a database user account, not just an APEX account, in order to connect to a database using SQL Developer. Second, it would be a huge security issue to expose any database to the public internet-- you'd realistically need to be on the internal Oracle network in order to connect to that database.
You could download and install Oracle XE on your local machine. That would let you develop an APEX application as well as allow you to connect to the XE database via SQL Developer (or any other tool you'd like)