Hi there,
1. I don’t think the problem is from the theme. I switched on the default WP theme and the problem remains (just 2 categories show). Also if you use the “Product Categories” widget in the sidebar the same 2 categories are displayed. Where did you get your import file from? Was it from an older version of WooCommerce?
2. If you check the demo page there is a search field but you need to set the header of your site like the one in the demo page.
3 & 4. That happens because you’re using the default post search of WordPress and not the product search. The default search of the theme display the results in columns just like the products page.
Cheers!