On Phone: Spotify Premium, otherwise PowerAmp.
No, you need to navigate to the playlist on your phone and just toggle it:
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If you add a song to the playlist on any device, it will sync/download the song on all devices that have it toggled. You can set it so it only downloads on WiFi. Green icon on left-hand side means that song has been downloaded.
Note playlists like daily mixes cannot be downloaded as they are dynamic, it will keep adding songs at the end based on what you liked/disliked. If you like the daily mix, just save it as a playlist.
Offline feature is only for premium, song radio (Probably spotify's best feature), the number of daily mixes, the non-forced shuffle on mobile, etc. are also premium.
Note that devices such as the Google home mini will only shuffle through the first 100 songs of the playlist, this does not apply to PC/phones.
On desktop:
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Toggled on my desktop:
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Can't really show it downloading as 100Mbps line, but green icon shows up for a second in playlist.
Meanwhile on my phone it's not downloaded:
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I'd need to use the toggle.
Most of the stuff is downloaded though as in my liked list.