Hello,
1) Go to Appearance > Widgets > add the filter widgets to the Shop page sidebar area. To be able to filter by size, material etc create appropriate attributes and add them to your products https://docs.woocommerce.com/document/managing-product-taxonomies/#section-6
2) It appears is you add the Product filter widget to sidebar.
3) What do you want to display there: tags, product categorie? Do you want to show that bar for all the pages or for the shop page only?
4) Nope, we don’t have such option. Look for a third-party plugin that will allow you to add custom badges.
5) We don’t have such option in our theme.
6) It’s possible to add custom elements on hover only by additional customization.
Regards