In the attempt to shorten the menu, I decided to put the lists of monthly archives and categories into drop down menus. I just realized that in time they will (at least the monthly archives) get longer and longer and longer…

So, my first problem was that I’ve made a hidden category for static pages and it would show on the menu if I used WP’s built-in dropdown_cats tag.

Luckily, Ria at Typed by has made a Drop Down Cats Plugin with ability to exclude. What’s nice about it is that it a) formats subcategories differently from parents (indented with a hyphen), and b) it includes a post count parameter.

Archive tag (wp_get_archives) has an option to format the list of archives as an option list. You just need to build a drop-down menu function around it. Thus, as found at the WordPress forums…
(remove the spaces after dots in “window.location=” if you’re going to cut-and-paste that — I needed to separate the words because it wouldn’t wrap nicely)

<form name="archiveform" action="">
  <select name="monthly_arch" onchange="window.location = (document. forms. archiveform. monthly_arch[document. forms. archiveform. monthly_arch. selectedIndex].value);">
    <option selected>Monthly archives</option>
    <?php wp_get_archives('type=monthly&format=option'); ?>
  </select>
</form>

It includes a little JavaScript, but I don’t know if that can be avoided. I’m disappointed to notice that formatting the archives as options is not compatible with showing the post count.
But
In the same thread there was a solution to that too. It involves a little tweaking (or correcting…) in template-functions-general.php, function get_archives_link around line 180, changing

else if ('option' == $format) {
return '<option value="'.$url.'">'.$text.'</option>'."\n";

to

else if ('option' == $format) {
return '<option value="'.$url.'">'.$text.$after.'</option>'."\n";