Is there a free software that allows to edit database ER diagrams and export for different SQL database systems (e.g. MySQL, Oracle, SQLite, Microsoft SQL Server, etc.)?
It won't export to everything you want, but I've had a lot of good experiences with Oracle SQL developer data modeller (Not to be confused with Oracle SQL developer). I currently don't have a copy running so no screenshots, but I found it significantly easier to use in designing a database ERD than visio, and forward engineering DDL from it. Once you've created the ERD for a database, you simply have to tell it to export the DDL (file -> export -> DDL) and it'll generate the commands for creating the database for you.
It runs on windows and linux, as long as java is installed. It will allow you to design an ERD and convert that to the necessary DDL for Oracle, DB2 or MSSQL. It dosen't support mysql or sqlite (Oracle has mysql workbench for mysql - but I have no direct experience with it)
SQL Power Architect: Data Modeling & Data Profiling Tool will allow you create ER diagrams and then forward engineer that diagram into DDL for Oracle 9i/10g, PostgreSQL, Microsoft SQL Server 2000, Microsoft SQL Server 2005, MySQL, IBM DB2, or HSQLDB (but not SQLite).
It won't allow you to edit an ERD created in a different program, although you can import / reverse engineer a database schema and edit that imported diagram.