Are expectations the root of unhappiness?
I think the "root" of unhappiness is (according to the second noble truth) meant to be various types of "craving" e.g. as described here.
Slightly more broadly, the three poisons are said to be at the "root" of the wheel of life.
Is being ambitious a bad thing?
I think the Buddhist concept called "Right Intention" could be described as a form of ambition ... and so can "Right Livelihood".
Also, there might be some important differences between Buddhist monks and Buddhist laity (see for example this answer): and part of the reason for that might be a difference in their view of ambition.
Can you be ambitious and not have expectations?
Buddhism is meant to be useful to different people who are at different stages of enlightenment: which is I think what is meant when it's described as, "good in the beginning, good in the middle, good in the end".
Also, I think we're supposed to remember that there is a Middle Way -- by which I mean that whenever someone asks the question "is it all or nothing?" then I think that a Buddhist answer is often, "it's neither" (or "it's both", or sometimes "that's the wrong question"). For example instead of "ambition: yes or no?" there are other questions to ask -- like "what should we be ambitious for?" and "what happens to us if ambition is thwarted?"
Alternatively, is it how we handle failure or disappointment when things don't go according to plan that is what creates unhappiness?
Yes, that too.
For me personally, the way I interpret it (which isn't how other people must interpret it) is that, if I have an expectation and if that expectation doesn't work out as planned, then my understanding of Buddhism suggests that I 'should' (and 'will') let go of (i.e. change) my expectation, instead of being attached to my previous and now-obviously-unskillful expectation.
FYI there's this piece of 'Zen' advice titled No Attachment to Dust which includes the line,
Live with cause and leave results to the great law of the universe.