IOS – How to stop next podcast always auto-playing when current one finishes in iOS

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.

The Podcasts app was updated for iOS 7 on October 22. I don’t think that this was a feature of iOS 6’s Podcast app, but if you are able to update to the latest version, there is an Edit button that can delete/mark-as-read multiple episodes or multiple subscriptions. (Assuming you’re not running an older device that isn’t compatible).

There isn’t a way to so multiple selects before a swipe to delete, though most apps have an “Edit” button to select multiple items before hitting the Delete button (as seen in Mail). In third-party apps, the interface depends on what the designer/developer built.