First of all Windows 7 is NOT supported for Framework manager ..
Cognos support should know that.. http://www-01.ibm.com/support/docview.wss?rs=3442&uid=swg27014110
This means it might
work but it has never been tested.
To clarify some points in your post:
1. JDBC driver is only required for the content manager component. The jdbc connection is used for the content store, metrics store(s) and audit database. When you query for example the audit database Cognos used ODBC or native client not jdbc!
2. When you develop in FM a model on a oracle database, you need the oracle database client (preferable same version as the database server) with the 32 bit libraries. When you test a query in you framework manager on your client the LOCAL oracle client software on your pc is used. After publishing to Cognos connection and running queries, the oracle client software on the server is used.
3. be sure that you use the 32 bit libraries and not the 64 bit. SQL plus can connect to the database with both libraries and Cognos with the 32 bits version only.
Try a workstation with a supported OS and see what happens...