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I've been learning to use block editor the past few days but cannot find a way to add images & icons to my header's dropdown menu like in the samples below. I've found online how to do it in the classic editor, but I am unable to switch back to classic editor. I would like to try it through block editor, but if there's a method to brute force it in classic editor and switch back, that's fine as well.

This is mostly depends on theme that you use, not block editor.
Try find icon setting in Appearance > Menus or Appearance > Customize > Menus
If there is no built-in option. You can add icon by using a bit of CSS or using some menu plugins.