WP Editor, save content and category on frontend

Use WordPress’s Ajax API. like this:

$.ajax({
    data: {
        action: 'save_post_frontend',
        content: // get your text
    },
    method: 'POST',
    url: ajaxurl,
    success: // on success
});

Now save the data in wp_ajax_save_post_frontend hook.
See documentation for more info.