Apps keep asking for Apple ID info after backup
You say the apps were downloaded under your friend's account, and they are paid.
Paid = You pay for it -- not friend pays for it.
There is a limitation in iOS preventing this from happening - you will need to know your friend's Apple ID and password to open the app - to prove it was you who bought them.
But why?
Without this limitation, there would be an epidemic of groups of college students using one Apple ID to 'split' the cost of apps, and have them on a lot of different people's devices. This is stealing from both the developer and Apple (Devs get 70% of app sales, Apple gets 30%), and Apple doesn't like to have it's cash stolen. Devs don't either. I'm one of them.