It is the correct hook. I often also add a bit of javascript, to active the checkboxes inside the form.
In your case it would not be necessary to use the $accepted_args parameter – 3 – of the add_filter() hook, because you’re not using any parameters, and simply work with a return value as integer, in seconds.
if ( ! function_exists( 'keep_me_logged_in_for_1_year' ) ) {
add_filter( 'auth_cookie_expiration', 'keep_me_logged_in_for_1_year' );
function keep_me_logged_in_for_1_year() {
return 31556926; // 1 year in seconds
}
add_filter( 'login_footer', 'set_default_true_on_checkbox' );
function set_default_true_on_checkbox() {
?>
<script type="text/javascript">
document.getElementById( 'rememberme' ).checked = true;
document.getElementById( 'wp-submit' ).focus();
</script>
<?php
}
}
If you also want to check for a closed browser, then you need a check via javascript, that js should also be added via the hook login_footer. See this how to to learn more about the idea behind it.