Equal

If you want a form on your site you should take a look at Gravity Forms – http://gravityforms.com

This does pretty much everything with forms!

The query_posts argument would need to look like this:

<?php query_posts( 'cat=-3&posts_per_page=3'); ?>

http://codex.ww.wp.xz.cn/Function_Reference/query_posts

Where to past it depends on your theme but I would go for category.php.

I would do this:

<?php if(in_category( array( 5,10 ) ) { // if the current post is in categories 5 or 10 ?>
    <?php if ( !is_user_logged_in() ) { // check whether the user is logged in ?>
       <p>You'll need to be a member to view this content.</p>
   <?php } else { // user is logged in so go ahead and show content ?>
      <?php the_content(); ?>
  <?php } ?>
<?php } ?>

Try what @alchymyth suggests. alchymyth is right in that you need to redefine your list type.

Therefore it is probably a call to <?php if(is_single() { ... } ?> that is causing the thumbnail to show on the single posts pages. This will probably be in index.php.

Yes that is correct:

<?php if ( is_user_logged_in() ) { ?>
  <!-- code here for what only logged in users can see -->
<?php } else { ?>
  <-- code here for what users that are not logged in see e.g. a login form -->
  <?php wp_login_form(); ?>
<?php } ?>

The archives.php is a template hierarchy file and not a page template. Therefore you would need to create your own page template.

http://codex.ww.wp.xz.cn/Template_Hierarchy

http://codex.ww.wp.xz.cn/Pages#Page_Templates

Who is your host? Do they advertise that they run PHP?