Re: show search form or link to search.php on home page
search_form() is in template-functions.php which is a core file. I just use windows search function to look inside files (you have to make a change in windows to do it) and just looked for probably something like “function search” and it found the code.
Anyhow, copying the code I pasted and putting it wherever you want the search bar to show up should work. Once you get it showing up, it’ll take up most of the screen, so open search_form.php and change the sizes.