How can I have individual podcast episode artwork show in Apple Podcasts?

I can't advise on Podcasts for iOS and you could ask in the forum for that. In iTunes when subscribing, you can have images displayed for individual episodes, but they have to be actually embedded in the media file. You can use an 'itunes:image' tag within the episode tags, but these only appear (and very small) in the Search results when they show individual episodes.

As you are using SoundCloud you may not have the ability to do this: I don't know whether if you add the image to an MP3 file's ID tags it will survive the transition into a Soundcloud file.

I don't know if there is a method for adding this tag using an iOS device. You can add an audio file to iTunes 'Songs' library on a Mac or PC (using iTunes, not the Library folder - never touch that) by dragging the file to the Songs listing, and then add the image using get Info, (command-i) then copy the file out again (and delete it from iTunes if required).

I'm afraid I can't answer that - I don't know anything about Spotify or whether it makes any use of the feed. You may have images in the 'itunes:image' tags which it's using. iTunes, and perhaps Apple Podcasts, uses the embedded image.