Hi,
First, in your custom product field, you should only enter a number. So no $. Otherwise, it won't be able to process it properly.
I think that's the main issue you have here.
Also, in the settings of the global cart fee plugin, I think you want to leave the "price" field empty since it's supposed to take the value from the custom field in the product, not the value in the plugin.
Finally, in the settings of the custom field, you can see display and email display settings. You want to turn them all off except the "backend form" one so that you can see the field when you edit the products.