1. You probably can’t. In most themes, the search results template content cannot be modified from the user interface, you have to edit the template file.
You can add this to the template file:
<?php echo do_shortcode('[searchform]'); ?>
to print out the shortcode. For more details, please ask theme support, as the exact details of the implementation depend on your theme.
2. It depends on the theme styles, please ask theme support for exact details.
3. Indexing reads through the post contents and creates an index for Relevanssi. The index tells Relevanssi which words are in which post. This does not affect the PageSpeed, because it’s not a constant process, it’s a small task that happens whenever a post is saved.