How can I use the WordPress Loop and Pagination in multiple instances but different scenarios throughout my site?

In your main loop file, check the $args. It filters the query. you can use the same query on other pages using your own options.

suppose, on another page, you want to fetch the posts of a different category. your query parameters may be like:

$args = array(
  'post_type' => 'post',
  'cat' => 'DIFFERENT_CATEGORY',
  'posts_per_page' => 2,
  'paged' => $paged
);

change the DIFFERENT_CATEGORY with specific category ID.

then you can pass these arguments to WP_Query as in your loop.php.

if your page has multiple such queries, you should use wp_reset_query before running further query. More her: http://codex.wordpress.org/Function_Reference/wp_reset_query

Does it help?