I’m confused. Is this the new issue or this is a detailed explanation of the previous one? If this is the new problem then it would better to create new topic because you mix up all the questions in one topic and it causes misunderstanding.
We have visited your site from mobile device and the categories widget is closed, products at the first 3 pages of the shop are the same for both versions EN and GE, we did not check all the pages because you have a lit of products.
So, please, could you provide screenshots (using Gyazo or Lightshot screenshot tool) and explain what old data are you talking about? Provide links to products that are marked as in-stock but they are out of stock.
And the main question is: does this issue appear with our theme only and you don’t have it with the default WooCommerce theme Storefront. Because keep in mind that we are not responsible for the WooCommerce, WPML or WordPress issues, we are theme support and can help you only if the issue is caused by our theme.
