IBM Power Systems run the IBM i operating system. This POWER processor-based servers also run AIX, IBM’s UNIX operating system, and Linux. The MySQL i DB2 Storage engine is only supported with IBM i.

Yes.

The new IBM® eServer i5 servers extend the IBM eServer iSeries™ family. They are the first servers in the industry based on the leading-edge IBM POWER5™ 64-bit microprocessor. Today’s eServer i5 servers give the flexibility to move from one generation of technology to another without disrupting a company’s business. IBM i5/OS™ Version 5 Release 3, the next generation of OS/400®, features support for multiple operating systems and application environments on a single, simplified platform. eServer i5 servers do more with less.

http://www.redbooks.ibm.com/redpieces/abstracts/ga195486.html

IBM i (previously known as i5/OS and OS/400) is an operating system. It is an integrated, easy to use environment that was designed by IBM for efficient, resilient business processing. DB2 is an integrated component within the i operating system. They cannot be decoupled. http://www-03.ibm.com/systems/i/

No, DB2 is a family of databases with certain similarities as well as differences. IBM DB2 for i is a distinct product within the DB2 familiy. DB2 for i is the database that runs on the IBM i operating system on IBM Power Systems.
http://www-03.ibm.com/systems/i/software/db2/index.html