If your objective is to do something without needing to pay, you’re probably out of luck, as finding a new server with good retention that’s free is pretty uncommon, at least in .us. But with that said, I’m familiar with most of the software you mention (except for FlexGet). In listing them together, you’re conflating three distinct tasks: the end-user cataloging/listing, the download indexing, and the actual downloading.
In the first category, you have Sonarr, Radarr, and CouchPotato (the latter of which is defunct), and other similar software. Sonarr acts somewhat as a DVR for TV shows–you tell it, for example, that you want Doctor Who. It first finds that show in a database to see which episodes are known, then scans your hard drive for episodes you already have. It then searches one or more indexers (configured by you) for episodes you don’t have, sends those to appropriate downloading software (again, configured by you). Once an episode is downloaded, Sonarr then moves it to your library, typically renames it to conform to some specified format, and optionally notifies your media server software (Plex/Emby/whatever). Radarr and CouchPotato work similarly, but for movies.
In the second category, indexers are generally third-party services (NZBHydra is more of a meta-indexer). They index torrents or NZBs, and give the end-user software (category 1) the information needed to tell the actual downloader (category 3) what to download. NZBHydra, as I said, is a meta-indexer–Sonarr (etc.) will send a search request to NZBHydra, whereupon NZBHydra will send that request to other indexers you’ve configured. It simplifies configuration inside Sonarr (etc.), and also gives you feedback on how productive your indexers are.
The third category is the software that actually downloads the files–this includes NZBGet and SABnzbd for NZBs (newsgroup files), and any number of torrent clients for torrents. All of the category 1 software works with both NZBs and torrents.
From what I can tell quickly from their website, FlexGet is also category 1 software, though it isn’t primarily operated using a web GUI as the others you name are.