Your question is confusing but I'll try.

Item 1

The $post variable is set before the template loads. It is preset to the first post in the Loop by the main query, so on single post/pages it matches the post/page that you are on and you can use it in the header to display your thumbnail.

Item 2

This is the most confusing of your questions. I am not really sure what you want. Again, the $post variable is set before the template loads. You could use that to display the thumbnail from the first post in the results in the header of the page. You would be using "the featured image for both posts and header" but there are potentially multiple thumbnails on a category archive so I am not sure that is what your really want.

Item 3

By default, there is not way to do this, but it pretty easy to have multiple featured images by way of custom meta fields.

See also: https://wordpress.stackexchange.com/a/109468/21376 A synopsis of that link: Trusting $post out of Loop context is a bit dicey unless you really understand WordPress template rendering internals. Using a function called get_queried_object can help get more expected behavior.