How can I extract and print specific field values from $page['content'] in page.tpl.php (vocabulary page)
There are a few things here:

Taxonomy term pages are not nodes, and on a taxonomy term page you are going to have the following args:


arg(0) ~ taxonomy 
arg(1) ~ term 
arg(2) ~ tid (taxonomy term ID)


To override term pages copy site_root/modules/taxonomy/taxonomy-term.tpl.php into your theme templates folder to override all taxonomy term pages. You will also find really helpful instructions in that template file.