Creating a Views Search Page AND Block (not just a search block and results page) I have a site with a box on the home page (only) that has a search box. I also have a 'search' link available on all pages of the site, in the navigation. The search link takes the user to a page called 'Search' where the search box appears (again) and allows them to do searches there. I've built the search for that page so the search box appears and the results display only after the user types in a search.

The main question I have is this: how do I create a block for the home page that ONLY displays the search box (results of such search still showing on the results page I've created in Views)?

I know how to expose the View as a block but this doesn't solve my issue because I want the search box itself to still show on the results page too.
Consider using a block with php filter to display the exposed search box in a block.

Example below is based on Search API with blocks exposed. I'm taking a guess you're using Search API based on your description.

For Search API:

$view = views_get_view('your_view_here');
$view->set_display('Page');
$view->init_handlers();
$exposed_form = $view->display_handler->get_plugin('exposed_form');
print $exposed_form->render_exposed_form(true);

For Drupal's Core Search:

$block = module_invoke('search', 'block_view');
print render($block['content']);