Could it be that I have not all the icons that are supposed to be in the JetMenu settings?
It can be sometimes that the available JetMenu item icons are displayed poorly in the Item Icon settings tab. Although they are all there, you cannot preview them to make a choice on which one to use.
To fix it up, please go to WP Dashboard > Elementor > Settings, click on the Advanced tab, and make sure Load Font Awesome 4 Support is set to “Yes”.
Then press the “Save Changes” button and go check out the icons. They should look like this now: