How do I remove old episodes?

Because of the way RSS feeds work, episodes cannot really be deleted.
Yes there's a Settings category called 'Automatic cleanup' that displays some deletion features, but it's only for downloaded episodes. Only the downloaded files are deleted.
If you don't want to see past episodes in the app, you have to mark them as 'played'. This can be done automatically after the episode as been fully listened to.
You can also do it manually (long press the episode => Mark Played) or you can long press an episode and select 'Mark older episodes as Played' in order to apply this flag to all the past episodes.
By default episodes marked as played have a green triangle with a checkbox inside displayed on top of them. Then you can press the toolbar 'Eye' icon in order to hide/show played content. You can do this in a screen displaying a list of episodes, but you can also do it in the app main screen. In that case you will only see podcasts having at least one unplayed episode.