Yes, the climb up the Duomo is included.
Yes, you have to book it.
At the Opera del Duomo office, you pass by and pick up the special pass for the Duomo monuments and book at that time.
Extra info: this is a recent change ...
There is now a requirement that all need to book the climb up the Duomo. So when you get to Florence, pick up your Firenze Card, then go by the Opera del Duomo office across the way from the and pick of the pass for the Duomo monuments. At that time, you can book a specific time for the climb up. It is likely you will find a time for later that day or the next, as everyone who doesn't have tickets before hand has to do the same! It is a recent change so it will take a while for most to even learn they have to do this. I just learned about it myself -- it wasn't on the Opera del Duomo website a few weeks ago when I last checked for a similar question.