-- Supports Searching

-- Can group subscriptions using tags

-- has got sleep timer (don't know about fading playback)

-- No ads

* Comprehensive online search/discovery for podcasts and individual episodes through the app

It does have a search for other podcasts, but could stand to be better. However, you can easily add an apple podcasts URL or RSS feed if you find the podcase elsewhere

* Group subscriptions into folders/tags/_something_ to help me organize the list

You can group subscriptions or episodes by playlists

* Sleep timer with fading playback

Pretty sure it can do this. You can also set default hours for sleep operation so if you wake up at 2am and want to listen to a podcast to get back to sleep you don't have to re-setup the sleep function.

* No in-app visual ads or other ad breaks that are not part of the podcast recording itself.

Not that I have noticed.