List post titles only when selecting a catagory

The simplest way: create a Category Template, e.g. by saving your archive.php template file as category.php – and edit out everything you don’t want there (like the_content etc.)

You probably have set your blog in admin > Options > Reading to show 5 posts. That applies to ALL your views (index, archives, catgeories, search etc.)

To have different number of posts on different views – you need to use a plugin:
http://mattread.com/projects/wp-plugins/custom-query-string-plugin/