Hello,
1- Actually for the Categories Menu dropdown effect on the Homepage, you can edit your Homepage with Gutenberg editor (i.e. simple edit) > in that you can find the Page Settings options > in that click on Header > there you can find the option of Open categories menu > you can disable that and the Categories Menu will work as collapsible like you want: https://snipboard.io/d1U6jQ.jpg
2- As far as we have checked this concern with the AJAX Search element added on the Homepage of your Site, it is appearing as Full Width on all viewports being added. Here is the following Video for reference in this regard: https://www.veed.io/view/96151de8-ac80-4e15-93e5-d5b84324ce8a
3- Actually, as you know the default Archive pages come from WooCommerce by default. So, you can’t edit them directly with the Page Builder plugins like Elementor or WPBakery.
But you can make use of the WoodMart Layout Builder to create Custom Archive pages layout including the Shop, Category, Single Product page, Cart and Checkout pages.
You can read more about it here: https://xtemos.com/docs-topic/woodmart-woocommerce-layout-builder/
Best Regards
.
