How does Apple's Podcasts app's Auto-Download of podcasts work? 1) In the Apple Podcasts app, one of the subscription options is Keep Most Recent Episode.  Is this the most recent unplayed episode or the most recent episode in the feed? If the former, is there any way to get the latter behavior? 

I subscribe to a couple daily news podcasts that I may not want to listen to every day. I just want the latest episode to be downloaded to my phone, so on the chance I want to listen, I'm listening to today's news and not last week's.

2) How often does it check for new episodes?  How do I ensure that it does check?  Sometimes I turn my phone off, and when I turn it back on again, the Podcasts app won't check for new ones until the next time I open it.  This seems weird; I would expect it to work more like the calendar which is running in the background syncing silently.

3) If 1 &amp; 2 are not possible, is there an alternative app you can recommend?  Can I use the iTunes app without syncing to my computer, for example?
I'm not sure about your first question. I suspect that Keep Most Recent refers to the most recent in the feed, but I don't know for sure.

As for the second question, the Podcasts app adheres to Apple's rules about background/scheduled downloading. That is to say, it only checks for new episodes when the app is open. This is a power and performance optimization decision, and for better or worse, no third party apps are allowed to schedule background checks either (although in theory they could on a jailbroken device, but I don't know if there are any apps that do this). Apps can trigger background downloads using geofencing however (the same way Reminders can trigger an alert when you leave your office).

There are several other podcast apps on the App Store, which have finer grained controls for subscriptions, and so forth. Downcast and Instacast are likely the two most popular ones.