Hi,
1) For the search module, you have to use the standard Joomla search module and enable the HikaShop product search plugin, edit the settings of this plugin in order to add the desired search columns (product_name, product_code, etc.)
Then the search will display the products with a link to the product page.
If you want a product listing shown when searching on a product you indeed have to use the filters.
2) This don't seem to be possible.
Joomla create a Cookie in the browser of the users when creating a session, it use this cookie to get the session infos on every pages.
So when you close the browser and open it again, as you still have the cookie, Joomla is able to get back the user session.
In HikaShop as we use the Joomla session, we are not able to know if the loaded session is the one before or after the browser has been closed.
What will be required is a system to delete the cookie created when closing the browser, but I don't think it's possible.