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Always enter and exit the US with your US passport. Not doing so is illegal.
For anything else, use your German passport, especially in the EU. That’s what I have been doing for years.
I don’t know if it is an issue for immigration, but you absolutely are required to file taxes every year and can get in trouble if you don’t. There’s a “streamlined procedure” that you can do if you’ve never filed taxes to start filing without incurring any fines or penalties.
Even if you’re downloading the file directly from the URL found in the RSS feed, that doesn’t mean that ads can’t be dynamically injected into the file. A URL like https://download.my.podcast/episode4.mp3 can still be answered by a script that serves a custom version of the podcast with region specific ads.