Alter search block form on subtheme

Iam trying to alter the search-block-form (edit tooltip message, add additional html elements and rearange it), but using the hook_form_FORM_ID_alter does not seem to be the appropriate way to do it. Printing dsm($form) does not list the attributes I am looking for (tooltip attribute).
The question is, where can I add/remove html elements and edit the tooltip of the search-block-form?