Ugly side module list style

Hi,

If you set the option "Show sub categories" to "no childs", the system won't add any styling and will use your template's style.
If you set this option to something else, the system adds some sliding CSS. In that case, you can edit the CSS easily in the configuration under the tab display. You can search for the CSS block with the comment /* sliders CSS for module categories display */ in order to find the CSS related to that part and then modify it to suit your needs.

In the module you need to set the "menu" option to a hikashop "listing of categories" or "listing of products" menu so that the system will know how to display the category thanks to the options of the menu.

Yes. The options of a module let you customize how the module itself is displayed. But you want to specify as well how the main page will look once you clicked on a link of that module. That's what the menu option is for. The option of the menu selected there will tell how the destination page will look. Of course, if you already have a menu configured, you could just use that menu in the module.
You can look at our demo website in the options of the categories module. There, the "listing of categories" menu which is on the top menu is used as reference menu in the options of the module so that when you click on the category links of the module, the page will display the same way as when you click on that "listing of categories" menu.