How can I POST or GET to the same admin page from which I am POST-ing or GET-ing

While WP is a little weak about working with forms in general, there are some loose conventions that can help keep things organized.

For POST forms there is a dedicated wp-admin/admin-post.php endpoint, which meant for just that and works similarly to better known admin-ajax.php.

Unless you have good reasons to organize it otherwise I would recommend just to use it for form processing. After you are done you can get back to your page with something like wp_safe_redirect( wp_get_referer() ); die;.