Q: Regarding a problem when I'm trying to enter iCloud in my new iPhone 5s

No and no. The new iPhone is already synced with the previous iCloud copy which I made just before changing to the new phone. It's not about setting up the iCloud account on the new phone either. It's about the fact that the phone needs to sync and therefore create an updated iCloud copy, but iCloud has reached it's max capacity. Although, when I'm trying to enter "storage settings" in the iCloud setting of the phone, I'm unable to enter it. Is it possible to send you a video of the actual problem?

Unfortunately yes... and while having the Charges pluged in as well. I'm lost here.