Can a black hole ever be the output of a Feynman diagram in momentum space? For a Feynman diagram representing a collision, particles come in from infinity with certain momenta, collide and then go off to infinite with other momenta. At collision vertices we integrate over all space-time.
It seems to me, that no in-coming or outgoing particles can be combined states if those states have any kind of half-life, because after infinite time they will decay. So even Feynman diagrams with protons incoming protons can only be approximate. (assuming proton decay)
Equally, if a black hole evaporates, can a black hole ever be one of the incoming or outgoing states?
As I read it, we could only have elementary particles as input and outputs, with perhaps a black hole forming during the collision which lasts for a certain amount of time (even a billion years) and then evaporating and elementary particles coming out. 
In other words, in quantum gravity, should the S-Matrix contain only information about colliding elementary particles or include colliding black holes?
 A: Let us be clear that any Feynman diagram of calculation implies quantization of gravity, for which there is no standard model yet.
That said, it is true that the standard model can be embedded in string theory models and also true that string theory  can accommodate quantization of gravity by including the  graviton in its string excitation.
Here micro black holes have been considered as a prediction of the extra dimensions of string theories, to be checked at LHC.

Such quantum black holes should decay emitting sprays of particles that could be seen by detectors at these facilities. A paper by Choptuik and Pretorius, published in 2010 in Physical Review Letters, presented a computer-generated proof that micro black holes must form from two colliding particles with sufficient energy, which might be allowable at the energies of the LHC if additional dimensions are present other than the customary four (three spatial, one temporal).

For example in this calculation they use semiclassical arguments to estimate the generation and decay of micro black holes at the LHC energies, based on the extra dimensions of string theories, thus connecting the standard model interactions with gravitational interactions.
The CMS experiment has been looking for them "Search for black holes and other new phenomena in high-multiplicity final states in proton-proton collisions"

Semiclassical black holes and string balls with masses as high as 9.5 TeV, and quantum black holes with masses as high as 9.0 TeV are excluded by this search in the context of models with extra dimensions, thus significantly extending limits set at a center-of-mass energy of 8 TeV with the LHC Run 1 data. 

Various string model predictions are tested. If you read the links, it is not simple Feynman diagrams, but extensions of such calculations within string theory models. In this sense colliding black holes can be accommodated.All this is highly speculative of course, as only just limits are given at the moment.
A: Note that Feynman diagrams are generically used for theories with weak coupling. As such, the "in" and "out" states must be "free fields" (the fields one gets when canonically quantising the field). This means essentially that Feynman diagrams capture "weak effects" since the S-matrix expansion only works perturbatively (such that the expansion makes sense).
Now, when one thinks of the S-matrix of gravity, they're thinking of the scattering of gravitons perturbatively. In practice this means that you have to consider small fluctuations around flat space ("weak effects").
To answer your question now, black holes are a strong effect effect of gravity and thus cannot be represented by calculations involving weak effects, such as Feynman diagrams. The in/out states must be free fields (such as quarks, photons, or even gravitons if naively quantised) and since black holes are interactions between gravitons (which are by no means weak), they cannot be represented by free states in a Fock space.
Scattering amplitudes in string theory make sense at high-energy whereas those in quantum field theory may break down. Essentially this is how one can capture black holes in string theory, because it can describe those strong effects. But Feynman diagrams are not present in string theory and one usually integrates over the worldsheet of a certain process.
I'll summarise by answering more directly your questions.


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*"Equally, if a black hole evaporates, can a black hole ever be one of the incoming or outgoing states?" 
No, since it is not a free state. Even in practical S-matrix calculations for protons one approximates by using the composite quarks (approximation: the quarks are treated as free (but actually are not), so one can use them as in/out states). Even a small amount of time from our perspective is to captured well by us assuming the states are defined at +/- infinity (this is why decay etc. is not a huge problem for the S-matrix).

*"in quantum gravity, should the S-Matrix contain only information about colliding elementary particles or include colliding black holes?"
In general the S-matrix should only contain information regarding the states of the Fock space. In the case of quantum gravity, this would be graviton states (plus other matter states if your theory contains more fields), which will be well-defined on the Fock space. So the S-matrix will not contain information about black holes, only gravitons + matter. In one could somehow include gravity into the standard model then the S-matrix would contain information only about the elementary particles and gravitons.
