Is there a file hidden somewhere in the internal storage that contains all the system setting preferences? I want to be able to access and modify this file (if it even exists) without rooting my device.
No, it is not possible to modify the Settings' app's files without root access. For the record, if you gain root access in future, then:
For Android 4.x-5.x
/data/data/com.android.providers.settings/databases/settings.db # database of Settings Providers/Storage app /data/data/com.android.settings/shared_prefs # some preferences saved by Settings app
For Android 6.0.x:
/data/system/users/0/ # look for the files with name beginning with settings_. They serve as the alternative of erstwhile settings.db /data/data/com.android.settings/shared_prefs
Files under shared_prefs and 0 (0 corresponds to User ID 0, the id of primary user) directories are human readable XML files so you can use a text editor to edit them on the fly. To edit settings.db, you can consider an app or processor's architecture (e.g. arm) compatible version of sqlite3.
Recommendation: while I've not tried the paid version but only the free version, the app My Android Tools claims to show and allows editing SharedPreferences as well as the SQLite databases of an app, so for Android 4.x-5.x the said app can be of good use.