1.) Depends on what you mean by using. If you mean more at a DBA level, then it will somewhat depend on what storage engine you're using with MySQL.
If you've mainly used MyISAM, then you'll have things like redo and undo storage segments to get used to when using Oracle as it is a fully transactional database.
Oracle also has a different and IMHO a more mature stored procedure language (PL/SQL) than that of MySQL. Oracle also incorporates more features than MySQL. Some examples are: function based indexes, index organized tables, better partitioning options, etc. (Some of these are on the table as future enhancements for MySQL)
2.) General SQL syntax for the most part is the same. Though the There are some differences in the way you directly call stored procedures. (i.e. call vs exec/execute)
3.) Oracle SQL Developer isn't bad and it's free. If you have a budget, Toad for Oracle is really nice. A web based application (PHP) that is useful is PHP MyAdmin, and the out of the box Oracle Enterprise Manager service is very useful.
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5.) I would recommend highly the Oracle documentation itself and My Oracle Support (formely MetaLink) http://metalink.oracle.com The Oracle Technology Network is also a good resource
6.) Oracle runs on all if not most major platforms including Linux, Windows, and AIX.