Q: Deleting podcasts

If you use automatic syncing with podcasts, check current settings on iPod's Podcasts settings screen in iTunes.  To do this, click the iPod's device button in iTunes.  The screenshot below shows upper left cover of my iTunes window, with iPod's device button (circled in red).

Click there, to show the iPod's Summary settings screen.  Along left side of window, in the sidebar, under Settings, click Podcasts.  The iPod's Podcasts settings screen appears to the right.  For automatic syncing (and not manual loading) of podcasts, checkmark the box for Sync Podcasts.  Set up the rest of screen as desired, to tell iTunes how to sync podcast episodes.  Click Apply.  Going forward, iTunes automatically syncs changes to your podcasts library when you connect iPod (or click Sync if already connected), based on how you set up that screen.

There may be data corruption on the iPod.  I would go to iPod's Summary settings screen and do a Restore of iPod using iTunes.  This erases the iPod, reinstalls its software, and sets it to default settings.  Then go to the iPod's Podcasts settings screen and set up automatic syncing for podcasts again.

I don't understand...  You said you have iTunes set up to automatically sync podcasts, so all podcasts on the iPod should be in your iTunes library.  With automatic syncing, items on iPod must also exist in iTunes library.  Doing a Restore erases the iPod, but you can just set up the iPod's Podcasts settings screen again, like it was before, and sync them back to the iPod.

If I'm not understanding your situation, please clarify.