First, whenever you use get or post variables, you should always use the vbulletin input cleaner to make them safe. If you don't know how, there's an article somewhere here.
Second, why would you want to get rid of the sessionhash? It's only showing if you have cookies disabled, and if you remove it, it will break login for people w/o cookies.
Third, as I said, when you change pages, you basically reload them. Post variables won't survive that. You need to save the get parameters into variables, pass those variables as parameters to the pagination URL, and then read them from the URL parameters upon reload.
Basically (forgoing input cleaner here, for simplicity:
PHP Code:
// use if isset constructions below to decide
// whether $_POST or $_GET should take precedence
$var1 = $_POST['var1']; // coming from your post form - would be easier to use get-method there, too
$var2 = $_GET['var1']; // coming from the URL parameter
// your code
// then, when initiating pagination add your variable as parameter
$pagenav = construct_page_nav(
$vbulletin->GPC['pagenumber'], $perpage,
$cel_ped['ped_count'], $vbulletin->options['bbr_peds_url'] . '/results.php?' . $vbulletin->session->vars['sessionurl'] . 'var1=' . $var1,