Can I add a search box to my homepage on its own, and show the posts table in the search results?
The search box provided in Posts Table Pro only ever appears above the table, and never on its own. When you type into it, the items in the table instantly update underneath in real time.
If you want to show a search box on a page without a table (e.g. on your homepage or sidebar), then you will need to use the default WordPress search instead. This creates a standalone search box that you can add anywhere on your site.
If your page contains a widgetized area then you can add the search widget there. If you need to add it to the main content area of a standard page, then you can use a plugin such as AMR Shortcode Any Widget which lets you put widgets anywhere on your site.
To show search results in a table layout, you will need to follow the instructions for adding the table to your WordPress search results template.