> For some obscure reason the document doesn't work at all with Firefox 

Note that mozilla/firefox is much stricter about mime types and http
headers than IE. I notice that you have given your stylesheet a .xslt
extension, what mime type is it hat servered with from your server?
It will need to be an XML one otherwise mozilla will not use it as a
stylesheet.
Also notice that although youhave a dtd reference mozilla/firefox never
loads an external dtd so if that dtd is needed to define entities or
attribute defaults, these will not work in mozilla.

In either case (mozilla or iE) the encodings specified by the server in
the http headers  overrule the encodings specified in <?xml ...

So your php code is writing
<?xml version="1.0" encoding="iso-8859-1"?>';
and sending the header header("Content-type: text/xml"); I don't really
know php, if that's the only header that is sent then you should be OK
as the default encoding for text/xml is iso-8859-1 so as long as your
characters really are in that encoding it should just work...

  Yet when I open the XML page in the browser (Safari on Mac Os X) the 
  accented vowels turn into strange symbols.

If these symbols include accented capital A's that's usually a sign that
the file is in utf8 but is being read as latin1, try manually puting the
browser in utf8 encoding (view/encoding menu in IE)

David