Assigning classes to elements through CSS

No, that isn’t possible with pure CSS.

Only with JavaScript:

// jQuery
$("h5").addClass("ui-corners-all");

// Pure JavaScript
var elements = document.getElementsByTagName("h5");
for (var i=0; i<elements.length; i++)
{
  var el = elements[i];      
  el.setAttribute( "class", el.getAttribute("class") + " ui-corners-all" );
}