I have been continuously having problems with my device being compromised, and even after a factory reset, the problem still persists.
If an Android device is not instructed after a factory data reset, will it automatically back up the old/original data?
A factory reset erases all data on the user data partition. On Android devices with default encryption this process is unreversible. Any backup has to be taken before a factory reset is performed.
If you have enabled device backup to Google cloud then of course if you use the same Google account and restore data from Google Cloud a malicious app could be reinstalled. But that would require that the app is available via Google Play Store at the time when you log in after factory reset. Google tries to identify malicious apps and remove them from Play Store but that does not mean Play Store is 100% malware free.
However that does not mean malware can not survive a factory reset on the device itself. For that it would have been placed on the system partition or another partition that is not modified by factory reset.
Depending on the phone such a persistent malware is likely or not. If you really think your device is compromised then there is only one solution:
Buy a new device from a trusted vendor, enable a new lock screen PIN with 5 or more digits and that nobody who knows you can guess (no birthday or PIn used elsewhere).
Use a new Google account with a new, secure and complex password. Do only install trusted apps from large known companies you trust.
And most important learn how your device behaves and what is normal. A lot of users suspect their devices to be malicious because they don't understand how Android works and misinterpret device behavior.
This site is a good place to learn how Android works, read it several times a week and learn from the questions and answers other user post or just browse older questions as long as they are not about an Android version too far away from your device.
