How to make each post/page have one and only one category. Categories could have radio box instead of checkbox next to each.
Actually, I wrote a plugin for exactly that.
In the plugin settings you define which taxonomies you'd like to apply this restriction to.
My prefered solution would be to use Advanced Custom Field with the Taxonomy field.
This way you can add a simple html select field for category, and in the same time hide the default wordpress categories widget.
From within the WordPress Admin, I haven't seen anyone modify category selection to force only 1 category. However, a good solution for you might be to use Gravity Forms. Using Gravity forms you can create a "front end posting solution" and in doing this you can set the category selection to use Radio Buttons instead of checkboxes. Alternatively you could also use a dropdown menu, again only allows 1 selection.
Another alternative is to modify the wp-admin area (which you will need to to in order to keep the WYSIWYG portion of the content editor as I don't think the front end solution provides this. I would suggest you remove the standard category meta box, very easily using code from this page: http://codex.wordpress.org/Function_Reference/remove_meta_box
And then create your own categories metabox using a dropdown value, which you can easily create from this page: http://codex.wordpress.org/Template_Tags/wp_dropdown_categories