App Store won't install apps I'm trying to re-download, I see the iCloud icon instead of INSTALL icon
Strictly speaking, the cloud/arrow button is the install button. You are missing buttons to acquire the app and add it to your account (either GET or PURCHASE). 

Try signing out of your iTunes Store account, restarting your phone (power off and on), and sign back into your iTunes Store account. That cloud button should still appear and allow you to add the app to your phone (it should turn into a loading spinner, and then an OPEN button).