Subcategories buttons

Simply install, activate the plugin, then use the shortcode [subcategories_buttons] to display, wherever you need.

Yes. Look for Setting-> Subcategories Buttons. There you can simply Exclude categories (one, or more), set the number of buttons in a row (1-6), set the color of border and border radius of button, select, if you would like to display category thumbnail, or check, if you want to hide categories with no product.

Nothing special. Categories, which have image, would display with image, the others without it. No problem.

The plugin is designed to use especially on product-archives. Selecting the categories are based on your product category tree (see Woocommerce-Products -> Product categories), not on your menu. You will always get back the first lower categories level from current category. If you are on main shop page, other static page, post, homepage, or CPT (Custom Post type), you will get back the very first level of categories. Oppositely – if you are on the lowest category level, you will get back nothing (there are no lower categories, which could be displayed).