For what its worth, I have noticed this/similar issue for some time. When I update an App directly from my iPhone the badge icon number decrements properly. However, if I update an App in iTunes and then sync with my iPhone the badge (on the iPhone) doesn't change until there is another update ready to be downloaded.
I just live with it. It is more of an annoyance (to me) than a serious problem. I keep expecting it to fixed with an iTunes/iOS update, but nothing yet. Its been acting like this for a VERY long time (years and various iTunes/iOS updates - even when I replaced my iPhone 4s with a 6s). I haven't tried singing out/in of the iTunes or App Store.
The "Find My Friends" and "Podcast" issue seems to be something else entirely. Perhaps a recent glitch on Apple's part since these are now built into the iOS. I read in a different forum that if the above behavior really bothers you (with those 2 Apps in iTunes) to simply delete those Apps in iTunes (NOT on your device - they can't be deleted there since they are part of the iOS). Again, I just live it for now.