Music player showing the wrong album art
I found the solution to this. As I said in my question, the problem is that all my songs had their Album Title set to "Unknown" or simply not set.

To fix this, you have to change the Album Title to something unique. Here's what I did to fix this (automatically, for all of my songs in a single click):


Download mp3tag.
Drag all of your songs into the program.
In the navigation bar, click Actions > Actions.
Create a new action with the following settings:

Name: Name it anything.
Create a new step.
Action type: Format value
Field: ALBUM
Format string: %artist% - %title%

Select all of your songs in the program.
In the navigation bar, click Actions and then click your newly created action.


That should fix it.