How much can be done with the title bar icon?

My plugin’s icon is black with no background, which makes it invisible in dark mode. I will add a white background, but I have some questions:

• is the plugin icon only set on the publishing page?
• is it possible to change the icon at runtime?
• is it possible to have NO icon? (my plugin’s icon can be typed with Unicode, so if omitting the icon would omit the space it takes up, I could just type the icon and it would look correct in both modes)