Hi,
1. I see 130 results with that search in three result pages. I couldn't find any missing product for the categories I checked.
Please be specific. Note also in the search, the products are not sorted by categories, but by "ordering" as per your menu item screenshot. But that value is for each category. So the results you see is first the ones that are first in each category, then second, etc up until 48 and then they are grouped by category. This leads to the products in the same categories being dispatched among the different result pages. Maybe that's why you thought they are missing ? To avoid that, I would recommend to change the ordering column to "product_name" so the products will be more grouped together per category.
2. That's a good point. We never tested that case and no one reported it so far.
It comes from the fact that you have the "filter" display setting turned off in your menu item and that we added 6 months ago the AJAX refresh of results of a search via the filters system. But when you access the menu item page via the filter module search with a search without results, the view doesn't display the main div of the listing necessary for the refresh but the filter module, knowing it is on the results page still load the results in AJAX, but just can't display them on the page as the area is missing.
To fix that, you can edit the file "listing" of the view "product" via the menu Display>Views and change the code:
$html = ob_get_clean();
if(!empty($html)) {
to:
$html = ob_get_clean();
if(!empty($html) || hikaInput::get()->getVar('hikashop_front_end_main',0)) {
We'll add that change on our end too.