The Events table template plugin simply does not work that way. The Calendar plugin does display each datetime as its own entry though.
The search probably isn’t working because the shortcode is being used within the do_shortcode function. If the shortcodes are used directly on within the page content, then the search will work.
So instead of breaking those out into 3 tables, you can do one table and all three dates will be shown with the applicable table row.
With regards to the calendar colors you’re going to need to limit the categories to one per event. When you have more than one category on the event you will not be able to rely on the colors working as expected.