(Why) did the Buddha teach rebirth? Was rebirth introduced in the First or Second Turnings, and why did the Buddha teach it? E.g. was it necessary to understand emptiness, causality, etc.? 
For example, I imagine that some "reasons why" might include:


Maybe it helps deemphasize a person's current lifetime, highlighting its impermanence, so that one does not cling to it or themselves.
On the other hand, for some, it seems to be a source of inspiration to persevere on the Path, for a "good rebirth".

Would any of the Buddhist doctrines, excluding karma and rebirth, fail to make sense without rebirth?
The Truth of Rebirth seems to be one of the larger points of dispute between participants on this site. See here and here for similar questions and answers. I guess this is not all that surprising to me since the site seems heavily represented by Western practitioners. In my experience it seems that westerners have (and I definitely include myself here) a particularly difficult time with the Truth of Rebirth as taught by the Buddha. Here is what Venerable Thanissaro Bhikkhu has to say about this:


  Yet as these Buddhist religions have come to the West, they have run
  into a barrier from modern Western culture: Of all the Buddha's
  teachings, rebirth has been one of the hardest for modern Westerners
  to accept. Part of this resistance comes from the fact that none of
  the dominant world-views of Western culture, religious or
  materialistic, contain anything corresponding to the idea of repeated
  rebirth. Plato taught it, but — aside from an esoteric fringe — few in
  the modern West have treated this side of his teaching as anything
  more than a myth.
  
  For people who have felt burned or repelled by the faith demands of
  Western religion, there is the added barrier that the teaching on
  rebirth is something that — for the unawakened — has to be taken on
  faith. They would prefer a Buddhism that makes no faith demands,
  focusing its attention solely on the benefits it can bring in this
  life.


There seem to be many approaches on this site:


Simply denying outright that the Buddha taught rebirth.
The Buddha taught the reality of rebirth as a (re)birth that takes place in this very life whenever self-views of "I" and "mine" occur. No "afterlife" concept of rebirth was taught or acknowledged by the Buddha.
Admitting he taught it, but insisting it was just skillful means.
Rebirth as a combination of metaphor and upaya.
Those who take rebirth at face value, but insist that mere experience can end.
Like above, but do not posit an end to this mere experience.
Those who think rebirth (generally) can be known incontrovertibly through reason alone.
Those who think rebirth requires a dose of faith to be known (generally).


And probably a number of other views that I've missed completely or misunderstood. Myself, I'm somewhere between 6, 7, and 8, I guess, but I struggle with this.

Was rebirth introduced in the First or Second Turnings, and why did the Buddha teach it?

To my mind, unquestionably it was taught and introduced. I think the Buddha taught it because it was/is a conventional truth and the Buddha taught the truth.

Would any of the Buddhist doctrines, excluding karma and rebirth, fail to make sense without rebirth?

Yes, I think there are obvious logical problems and inconsistencies that present if you deny the truth of rebirth. For instance, as BodhiWalker mentions above one could seek the end of suffering by merely ending this current life. Even more problematic, one could "liberate" others from suffering by murdering sentient beings painlessly. That's monstrous and I want nothing to do with it. I want to make quite clear, I'm sure this is not what others have in mind when denying or glossing the truth of rebirth. I'm emphatically not concluding that those who deny or gloss the truth of rebirth are murderous monsters, just that denying or glossing the truth of rebirth presents logical problems that I don't see a way around.

Anyway, I think Venerable Thanissaro Bhikkhu's article above is excellent and highly recommend reading it many times and contemplating these questions.