Is $site_name , $logo ,$site_slogan are part of render array $page? is $site_name , $logo  ,$site_slogan variables values are store in render array $page? or they are independent from $page? if they are independent.then list of variables that have same nature?
The variables you use in templates aren't really part of the render arrays themselves.  They are created by the various preprocess and process functions and added to the $variables array.  By default, those you list are added in either a template_preprocess_page() or template_preprocess_html(), which your theme can have, too.  Look at the source to see what is going on.  For example, the stock template_preprocess_page sets $site_slogan like

$variables['site_slogan'] = (theme_get_setting('toggle_slogan') ?
                            filter_xss_admin(variable_get('site_slogan', '')) : '');


To aid in figuring this out, I suggest installing the Devel module, and looking at the functions that it provides.  For example, you can do

dpr($variables);


to get a dump of everything in the $variables array, or you can do

kpr($page);


to poke through the $page object via Krumo, which is included in Devel.