Hello,
This manual describes and contains all the options you need: https://xtemos.com/docs-topic/adding-menu-to-the-site/
2) I need good looking search box.
This manual shows how to add the icons and images: https://xtemos.com/docs-topic/product-category-menu/
3) Menu tab title should look minimal. (Like example) These tabs are not editable, unfortunately.
4) Texts should be Capitalize.
You can change the font by means of the Advanced typography option, which allows you to choose the item from a drop-down or insert your custom CSS class. Please navigate to the Theme Settings > Typography > Advanced Typography.
Here is the documentation providing more detailed instructions: https://xtemos.com/docs/woodmart/advanced-typography-settings/
If you have any questions please feel free to contact us.
Best Regards