IOS – Podcast App Double Speed Missing

From the Podcasts view (left tab), press the Edit button at the top left (you may need to scroll up to see it). Each podcast will have an (x) button in its top left corner. Hit that button (it's small and can be tricky, but it works) to remove the podcast. Use the Done button at the top left when you're finished editing.

If you run enough other apps for Podcasts to be unloaded from memory, resuming from the lock screen will start playing the Music app. There's no way around it. The only thing you can do for now is to delete the Podcasts app so you can play from Music again (after quitting and restarting it), and join the thousands of other people giving it 1 star in iTunes.