Suggestions on how to configure the settings for podcasts
I suggest starting over -- that is 

1) unsubscribe from all podcasts, and restart all devices. Then,
2) do Settings -&gt; Podcasts -&gt; Sync Podcasts and/or Preferences =&gt; Store =&gt; Sync Playback as described in the notes below your question. Then,
3) Plug in each device to your computer with iTunes open, select the device as shown in the first image, go to the `Summary` section for that device and make sure `Sync with this iPhone over Wi-Fi` is checked.




4) Finally, subscribe one at a time just from one iOS device only
5) After the above is working, subscribe from the other devices


I say this first because I've been subscribing to podcasts using iTunes on my Mac and my iPhone for about a decade without problems like you describe, and in the last several years I never even go to iTunes on my computer anymore.  I just subscribe to new Podcasts from my iPhone and things sync fine. Second, my experience is "starting fresh" tends to solve most computer-related problems.