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. 

1. Select + Delete key, or right-click, then select Delete.

2. You can subscribe to different podcasts on different devices. They will sync only if you subscribe to each podcast on both, iPod and in iTunes.

Are you syncing over Wi-Fi (iPod must be powered) or over USB cable? How often do you sync?

(1) I do not use that method, so I don't know...  When you use automatic syncing, you can set it up to automatically add unplayed podcast episodes to iPod and remove played podcast episodes from iPod (when you sync).  OR, you delete podcast episodes you no longer want from your iTunes Podcasts library (not from iPod); the next time you connect iPod to sync, iTunes automatically removes those episodes from iPod.

(2) If you have podcast episodes on iPod that are not in your iTunes library, you must have been using the manual method previously.  With automatic syncing, each time you sync, any items on iPod that are no longer in your iTunes library are removed from iPod.  OR, there is data corruption on the iPod that should be resolved by doing a Restore.

By design, iTunes syncing is generally in one direction only, from iTunes library to iPod.  You cannot sync podcast episodes from iPod to computer using iTunes.  It may be possible using a third-party utility, but they usually handle songs.  Podcast episodes are free, so people don't usually care.  You should see if you can just download them again, from iTunes Store.