Podcasts connect “Can’t parse your feed”

The podcasts part of iTunes just manage those you have either subscribed to, or subscribed and unsubscribed but still have episodes. It looks like your issue is related to trying to manually manage these, which will let you add the files to your iPod, but iTunes really doesn't know about the structure behind them.

I would recommend having it manage them for you. In that case, you can also tell it to delete those (or keep those) you want after you listen.

For example, I have a couple of concerts from NPR on my iPod that I keep, but my other daily/weekly podcasts delete after they have been listened too.

I expect the reason you are seeing the difference between iTunes and the iPod is the way you are currently mixing the two methods.

The feed URL can only be edited if Google Reader sync is off.

Select the feed in the left column and press ⌘I to bring up the feed info window. Click on the triangle next to the URLs entry and you will get a text box that will allow you to edit the feed URL. There are a bunch of other setting there that allow you to do things like specify how often a particular feed is updated, how long articles are kept, etc.