How can I add a home link to my websites menu and how can I remove the search box from the menu?
Solution - 1
If you provide the code that you have in your template it will be easier to help you...
Solution - 2
Menu codes are complicated. It can generated by wp_list_pages or wp_page_menu or wp_nav_menu. So you need to post your menu code here or at least tell people what theme your are using, so that people here can give you correct answer.
However a simple menu home link can be
Solution - 3
You have to edit your theme's header.php file. it should be like
in the same file, you will get something like this:
You can remove that to remove the search box