Is there any way to lock the Lock/Unlock settings? I've got an iPad mini 4, running iOS 10.3.3.

When you head to the Settings > Display &amp; Brightness, you can set if you want the iPad to lock when you close the cover/press the power button, as well as set a time for when it automatically locks. This does not, however, require a password.

Is there any way to lock these settings so that a certain person can't keep changing them? There doesn't seem to be a way to do this is the Restrictions or anything, and I can't find anything on how to do this.
No, unfortunately there is no way to lock this particular setting. 

This is similar to how it isn't possible to lock the auto lock setting on iPhones – if someone has access to the device in an unlocked state, that person can change the auto lock duration freely from the settings app.

However, you can use a device management profile from something like SimpleMDM. There are many other apps which allow you to hide or restrict system apps, like the settings app. In order to access the settings app, you will need to disable this profile which will require your passcode or your fingerprint. In this way, someone will be unable to modify this setting without your passcode or Touch ID authentication.