How do I redirect 404's to index.html and rewrite the URL to the home page URL?

You can use these 2 lines at the top of your .htaccess:

DirectoryIndex index.html
ErrorDocument 404 http://domain.com/

DirectoryIndex will make http://domain.com/ load http://domain.com/index.html by default and use of http:// in ErrorDocument will make it redirect to new URL.

Try below code :

RewriteEngine on
RewriteCond %{REQUEST_FILENAME} !-f
RewriteRule . / [L,R=301]

ErrorDocument 404 /index.php

It's any 404 url to your home page.