Is there a podcast app that can download podcasts so that they're available to the stock music player I'm looking for a podcast app that will just run in the background, download new episodes when necessary and make these episodes available to the default music player.

I like to access my music and my podcasts in the same place and don't really like the idea of using separate apps to listen to podcasts and the rest of my music.
The BeyondPod podcast player will download new podcasts directly to your device, on a schedule. For example, I have it run every morning to look for new podcasts.  And, it can be configured to "Share Podcasts" with the default Android Music Player.

I have never used the share-with-default-player feature, so I can't really comment on how well that works in practice.  But the basic podcast management and download features have all worked nicely for me.