Hi,
In the search plugins options, you have the option itemid that you can set to the itemid of your menu. That will be added to the url of the links in order to keep your modules on the page after clicking on the link.
For the fact that the results open a new page we will look at adding an option to select the behavior. In the mean time, you can change the line:
$select = ' a.product_id AS id, a.product_name AS title, a.product_created AS created , a.product_description AS text, "1" AS browsernav';
to:
$select = ' a.product_id AS id, a.product_name AS title, a.product_created AS created , a.product_description AS text, "2" AS browsernav';
in the file plugins/search/hikashop_products.php near line 110 and the links won't open in a new page.