Pass variable from action back to template

The difference between add_action() and add_filter() is semantic not technical, except that a filter expects a returned value and an action does/will not.
The question is whether contact_form_send_email() needs to be called-at /hooked-to init.

If not, you can define your filter in the template with apply_filters(), and then hook to it to run your function and return a value to it.

In template page:

$some_variable="some value";
$some_variable = apply_filters('my_filter_hook', $some_variable);

In functions.php

add_filter('my_filter_hook', 'contact_form_send_email', 10, 1 );

function contact_form_send_email( $some_variable ) {
    //do stuff
    $some_variable="some new value";

    return $some_variable;

}

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To help determine where you need to hook, this Actions Reference can be useful. Scroll down and you will see some template specific hooks as well.