App Store ID & a different iCloud ID?
There are instructions on how to set this up on Apple's support site, see towards the end of https://support.apple.com/HT204053.

Basically you open Settings on your iOS device and


configure/use one AppleID to log into iCloud 
configure/use the other ID to log into iTunes &amp; App Store


Having said that, I would still recommend to address the issues you have with your current account. To verify your billing details, log into the Store and update your information (if it doesn't work in the iOS device, you can use any computer running iTunes for this, just don't forget to log out within iTunes afterwards); to recover your second account, start with https://iforgot.apple.com/.