if home button cable has a tear causing touch ID issue, replacing it in a new display is not going to help. The answer to your question about display, any quality display for iphone 7 will work. If you really want to fix touch ID, you would have to repair the cable, which is not simple. You can get displays on ebay, union repair, mobile sentrix, rewa, etc…
typically the backups are done automatically by default. So that means, that you already have a backup in the icloud, but you cannot see it by logging into icloud (wisdom of apple). However, if you are activating (or resetting) a new phone, you will get an option of restoring from your latest backup, which was taken automatically. It will give you date and time to select from. I would recommend to do this before you are resetting your device and loose your files.
over the years, I did have a few ghost type issues after screen replacement with after market screens on iphone 7 that is. It turned out that the only solution worked for me is to replace the screen. I do believe that when the after market screen is matching to the OEM screen, that is when the problem is going away. So you would have to match the lg to lg or toshiba to toshiba etc… It is a rare problem, but it does happen and then you should pay attention.
I would try another screen and see if the same condition occurs to eliminate the possibility of screen, which may not be perfect match. I would also put the old screen back and see if that behavior repeats.