<p>What is the default XSLT processor for IE6. Does it support EXSLT? Is there anyway to make it support it?</p>

You have awakened a tormentor from my past.
MSXML, it was installed as a service update, and the XSLT processor originally included in MSIE6 did not support much of anything other than a very general level of draft functionality.  The service update improved on it greatly.
Because of requiring the presence of a service update we never considered performing any client-side XSLT with MSIE6 and instead performed transformations server-side when the client browser was MSIE6 and relayed the transformed content.