I had the same exact problem (on my 5c) and this fixed it for me.
Settings... General... Reset... Reset all settings
Then...
General... Restrictions...
ITunes Store "Enabled"
Installing Apps "Enabled"
Deleting Apps "Enabled"
In-App Purchases "Disabled"
General... Restrictions... Password Settings...
... Purchases & In-App Purchases...
Required after 15 Minutes CHECKED
... Free Downloads...
Require Password "ENABLED"
Then I went to App Store updates and updated all my apps (yes, I did have to enter my password this one time)
About an hour or so later there were a few new app updates, I pressed "update all" and they updated without me needing to enter my password !!
Out of curiosity I then went back into General... Restrictions... Password Settings... and nothing was checked
Free Downloads... Require Password is enabled but greyed out (and I can't change it)
I don't know if it was the settings, the reset, or both that worked for me (or if this will work for you) but it's worth trying... just don't forget to re-enable/login into your wi-fi after you "reset all settings" so you don't use up data.