I feel i'm missing out on something here, I have an Iphone and good headphones:

Do I need to download anything, in order to play podcasts? No you'll have a podcast app on your phone. But I like using spotify personally

Do I then use whichever app to download the podcasts? You don't have to download the podcasts they can play as a stream

Do they delete after a period of time or do you need to delete them? Depends on your settings

Can you part listen (ie pause and pick up again at that point)? Yes if you have a iphone it'll remember where you left off

Any recommendations for good podcasts? Loads but my personal recommendation is the Socially Distant Sports Bar - Elis James Mike Bubbins and Steff Guerro. You'll laugh for hours

Thanks in anticipation of any help.