Why do I have to get permission to save to my own phone? I am the owner of a new Samsung Galaxy J7 (J700T). When I try to delete files with no data on them on my internal, external, or SD card, the phone tells me I am denied. 

I want the files I've created. I would love to have this phone checked for spyware, but I am a Windows desktop guy and this Android flashing/rooting stuff has me stumped. Please send me in the right direction.

How do I get my permissions to stay in my favor? How does this happen, why I don't have the authority to do what?
As @Robert mentions in his excellent comment, the permission system on Android is not user-based, but app-based. Although newer Android versions have some multi-user support, the general attitude of the privilege system is that you, as the owner of the phones, have the privilege to modify the access of the apps. It is because the only human using the phone is its owner, so they can be trusted, and they is the one who should be protected from the generally distrusted apps. Before the mobile world, the case was its opposite: the apps was (roughly) more trustable, but the users using them were not.

With the default settings, in general, if you give an app access to your files (photos, documents, contacts and so on), it should work. A general "permission denied" for everything is clearly a problem. If your device is not rooted, the warranty now lives and you don't want to root / reflash it, then you might look for your warranty options.

Before that, I would suggest to try a factory reset. It might help. Considering your problem, you probably don't have data to lose with it. :-)