How does Apple's Podcasts app's Auto-Download of podcasts work?
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.