WordPress search: pretty URLs (permalinks), custom post types and pagination

I have three separate search boxes on my website: species CPT search, glossary CPT search and a general search which I’d like to search post and species post types.

My questions are threefold,

1. What’s the most efficient/effective code to make pretty search URLs, and is there a reason that WordPress doesn’t activate these by default?
2. Bearing #1 in mind, what code do I need to use in searchform-species, searchform-glossary, search and searchresults to specify that my species search will only return species results, glossary will return glossary results, etc.
3. How do I incorporate pagination?

The code I’m currently attempting is as follows:

CODE FOR #1

add_action( 'template_redirect', 'my_rewrite' );
function my_rewrite() {

    if ( is_search() and false === strpos( $_SERVER['REQUEST_URI'], '/search/' ) ) {
        wp_redirect( get_bloginfo( 'home' ) . '/search/' . str_replace( ' ', '+', str_replace( '%20', '+', get_query_var( 's' ) ) ) );
        exit();
    }

}

CODE FOR #2 & #3

searchform.php

<?php
    $search_query = get_search_query();
    if (!$search_query) {
        $search_query = "SEARCH";
    }
?>
<form id="topbar_search" action="<?php echo home_url( '/' ); ?>" method="post">
    <p>SEARCH</p>
    <input type="text" size="50" value="<?php echo $search_query; ?>" name="s" id="s" class="topbar_longinput default-value" />
    <input type="submit" value="GO" class="topbar_submit" />
    <p class="tinylinks"><a href="#1">advanced search</a></p>
</form>

searchform-species.php

<h1 class="profilesearch">PROFILE<span class="white">SEARCH</span></h1>
<form id="profilesearch" action="<?php echo home_url( '/' ); ?>" method="post">
    <input type="hidden" name="type" value="profile" />

    <input type="text" size="50" class="default-value" value="SEARCH" name="s" />
    <input type="submit" value="GO" class="profilesearch_submit" />
    <!-- <p class="tinysearch"><a href="/dev/advanced-search/">ADVANCED SEARCH</a></p> -->
    <input type="checkbox" name="showthumbnails" id="showthumbnails" class="checkbox" <?php if ($_POST["showthumbnails"] == "on") { echo 'checked="checked" '; } ?>/><label for="showthumbnails">HIDE THUMBNAILS</label>
</form>

search.php

    if (isset($_REQUEST["type"])) {
        switch ($_REQUEST["type"]) {
            case "profile" :
                $post_type = "species";
                break;
            case "glossary" :
                $post_type = "glossary";
                break;
            default :
                $post_type = array( 'post', 'species' );
                break;
        }
    } else {
        $post_type = array( 'post', 'species' );
    }

    $paged = (get_query_var('paged')) ? get_query_var('paged') : 1;

    $args = array(
        's' => $search_term,
        'post_type' => $post_type,
        'paged' => $paged
    );
?>

<?php get_template_part('searchresults'); ?>

What’s currently happening is this:

The individual searches work, but the pagination doesn’t. When I click my next/previous_posts_link button, it adds /page/2/ to the URL and does nothing. If I try to echo out the get_query_var('page') it returns NULL. If I try to echo out 'paged' instead, it returns 0.

Additionally when a user clicks this link, the $search_query variable from searchform.php is displaying as page/bettas (where bettas is the original query).

Ideally what I’d like to happen is this:

1. In the “general search”, a user enters betta. The URL changes to /search/betta which lists any post or species content containing the word betta.
2. In the “species search”, a user enters betta. The URL changes to /search/betta/species (or whatever is best for SEO) and displays a list of only species content containing the word betta.
3. The same user as #2 then clicks the Next button (previous_posts_link). The URL changes to /search/betta/species/page/2 and the user gets to see the next 25 species listed.

I’m a complete novice at WP rewriting and pagination so any assistance would be greatly appreciated.