This is the area where jQuery rules the roost. Or so everything I've read tells me.  So, how does Dojo stack up?

Still running this in parallel with Kyle's series:

1. Setting the contents of an element

   // Convenience function to hide query details
   var element = dojo.byId('whatever');
   element.innerHTML = "Change to this";

   Or you could chain it jQuery style:

   dojo.byId('whatever').innerHTML = "Change to this";

   Really just a matter of personal preference and what you find more readable.
2. Adding a new html element

   var l = dojo.byId("list");
   var li = document.createElement("li");
   li.innerHTML="Three";
   l.appendChild(li);

3. Removing content

   li = dojo.byId("garbage");
   dojo._destroyElement(li);

4. Setting css properties (actually, there are a ton of helper functions for setting individual styles, but here's the generic case):

   node = dojo.byId("whatever");  // or use dojo.query
   dojo.style(node, "color", "salmon");
   

5. Iterating over the matched set
   When you use a dojo query (using CSS3 selectors), you can manipulate all the results automatically:

   l = dojo.query("li");
   l.style("border", "medium double black");

jQuery wins this one hands down.  But you probably wouldn't be doing much of this sort of thing with dojo anyway.  You'd be too busy working with higher level abstractions.