If I download an app onto my iPad Will it automatically download onto my iPhone

[iMore Question]

The reason for my question is that I just purchased an iPhone 7 and previously had an iPhone 4. I also have an iPad. When I signed into my apple account on the iPhone 7 it asked me if I wanted to link my account with my iPhone 4 or my iPad. I linked it to my iphone4. Now I keep getting a message stating that my iCloud is full and I need to upgrade my memory. So I checked on my iCloud data usage and it has 3 devices saved with basically the same data on all 3. First of all shouldn't the info on my old phone be deleted because it was all transferred to my new phone? So technically there should only be 2 devices in my iCloud. Also, shouldn't they all be linked into one account with one backup for all 3? Meaning all apps, photos, email, etc. should be the same in all 3 and backed up in one file with iCloud, right? How do I sink all of my accounts into one so that the storage isn't backing up duplicates? Hope this makes sense. Thank you.