Podcast app which supports old episodes well? I am looking for a podcast player app which will let me subscribe to a podcast and listen to the earliest episodes first, eventually catching up to the current episodes.

I've tried Google Listen and PodCat, and neither one seemed to handle this case well. I don't want to install a paid app and start a 15-minute-trial until I know what to expect from it.

It should also be reliable at remembering the current position in a paused-and-ignored-for-a-while podcast, and download/delete automatically so I generally don't have to manage it except to actually listen.
BeyondPod ($7 but well worth it) will do everything you mentioned in your question.  I have tried DoggCatcher and a slew of free apps and I've found BeyondPod to be the best.  

You sort you podcasts from newest to oldest, or oldest to newest, or longest to shortest, or shortest to longest.  You can also break them up into categories which I find helpful.

Also, it is very good at remembering you place.  I regularly start one podcast then switch to another, then another and find that I can always return to the unfinished shows without losing my place.

There are also numerous autoupdate and auto delete options.  You can set certain categories of feeds update on a certain schedule.  You can set the auto delete globally, on the category level or at the podcast level, telling it to delete podcasts of a certain age, or tell it to only keep so many podcasts, or delete after listening.  Its also easy to mass delete all "listened to" podcasts with just two clicks.