I don't use the stock app, but does the badge icon show new episodes? That is how Pocket Casts works, so you don't need notifications on, but if you only listen to that one podcast maybe you wouldn't justify buying an app when you already have one that works.
You can, if you download them to a computer but how they are distributed may affect that. Many show have a website with a link to an MP3 you can download, and the SuperDrive burns just fine.
What do you mean by distributed? I went ahead & made a playlist & drag & dropped the podcast into there & burned it as a MP3. Is that what you would do