I found these tutorials the have exercises on the most common DOM methods, and have some exercises with Events and Ajax:
http://dom-tutorials.appspot.com/static/index.html
(The exercises at the end of each seem time-intensive to me. I recommend skipping them on the first pass, and if you want to, go back and to them later.)
I am happy to help when you have any questions.
Nerdy stuff about the DOM that you don't strictly need to know can be found in this article:
https://developer.mozilla.org/en-US/docs/DOM/DOM_Reference/Introduction
Something they mention in the article is that while the DOM and JS are two different things, in the very beginning, they weren’t. After all, JavaScript was invented as a way to give web page creators the ability to do all the things that the DOM allows. That is part of why they seem so tightly related; they are like siamese twins that got separated :)