I've used Tivoli with Oracle/Rman for several years now. It was a bit flaky when I first started using it (8i - ~2000), but it's MUCH better now. I don't know how I'd do my backups without it since I don't have that much disk.

What versions are you using?? TDP 5.2 works very well with 9i.

To answer your questions:
1) Yes it necessary to schedule a backup to delete jobs. TSM does not have control over them, RMAN does. But it's pretty simple. Set the persistent configuration for your retention policy:
Code:
CONFIGURE RETENTION POLICY TO RECOVERY WINDOW OF 30 DAYS;
Then run a job periodically to delete the backups
Code:
${ORACLE_HOME}/bin/rman  << EOF
  connect target 
  connect catalog rman/${RMAN_PASS}@RCAT
  delete noprompt obsolete;
EOF
2&3) There are multiple reasons the piece could be lost. It could be that the database was unregistered/registered. The first thing to try is to do a crosscheck. Make sure your are on the latest release for this. In older versions of Tivoli, crosscheck couldn't find ANYTHING on tape, and would delete all your backups from your repository... not pretty. I know it works in 5.2
Code:
${ORACLE_HOME}/bin/rman  << EOF
  connect target 
  connect catalog rman/${RMAN_PASS}@RCAT
  crosscheck backup completed before 'sysdate-30';
EOF
If that doesn't work, and the files are still on Tivoli, you can delete them using tdposync. Read up on the documentation on this(Tivoli Data Protection for Oracle Installation & Users Guide - Chapter 5 Using the Utilities) . It WILL delete old files on Tivoli that aren't in the catalog anymore. But be careful using it.

Hope that helps!
Jodie