Deleting Apps on iphone for good?

So I went through a deleted a lot of apps and songs off my phone/itunes, but it's like they're all still there… There's a little cloud icon with an arrow pointing down, which I thought meant that the apps/songs are backed up on my cloud. But my cloud isn't even enabled on my phone… How do I delete these things for good and not have them available for quick re-downloading from cloud, if there's nothing to be removed from cloud?

Added (1). My iCloud is not even enabled on my phone.
So I can't erase things backed up on my iCloud, if there's nothing there to erase.

Go into your settings and click icloud.pretty straight forward from there

You can just connect your i phone to your laptop, open your iTunes >> menu>> iphone>>musics>>enable sync music>>revert.

songs will be removed from your i phone, the same you can do for app.

That is the download icon, the items are not in your phone. You are most likely in the purchased section of the store.