In short no – the template used is based on the query, and when paginating you are essentially repeating the same query, but for a different page. In general the template will be the same.
archive.php can be used for most queries, but often preferable templates exist (e.g. category templates, tag templates, author templates etc.). What is preferable is determined by the template hierarchy. For the home page,however, index.php is often used, and the template will persist with pagination.
As for displaying only something on ‘page 1’, you can try the following:
<?php
//initialize $do_not_duplicate
$do_not_duplicate=0;
//is_paged returns true if we are on page 2,3,...
if(!is_paged()):
//Get featured content for page 1
$my_query = new WP_Query('category_name=featured&posts_per_page=1');
while ($my_query->have_posts()) : $my_query->the_post();
$do_not_duplicate = $post->ID;
//Do stuff...
endwhile;
endif;
//Display rest of content
if (have_posts()) :
while (have_posts()) : the_post();
if( $post->ID == $do_not_duplicate ) continue;
//Do stuff
endwhile;
endif;
?>
Not tested