As to solution #2, currently it is unfalsifiable due to the dark matter remaining at this stage at least a purely hypothetical object.

No, cold dark matter is a falsifiable theory. (And some even claim it is already falsified.) This is because CDM is a very special very particular model of imaginable dark matter. Namely, it is simply a massive particle, with some mass. Which is the only unknown parameter of the theory. Everything else follows from the equations. Of course, as for usual matter, observations have to be used to find the distribution of the dark matter. This has been done, by using their gravitational field.

Note: The gravitational field has ten components. The cold dark matter is dust, thus, is defined by the distribution of density and velocity.

Does GR predict the existence of dark matter?

Does it make predictions about gravity with out dark matter?

What does your answer suggest about GR?

No. GR does not predict anything about matter, neither of dark matter nor of visible matter. It describes the gravitational field, and how it interacts with matter. The theory of matter is separate.

Without adding a theory of matter, GR is not really complete. The full Lagrangian - which is what you need to get the full equations - consists of the GR Lagrangian (Hilbert Lagrangian) and a matter Lagrangian which depends on the theory of matter. If you find a new particle in the particle accelerator or so, you have to add a new term to the Lagrangian of the Standard Model of particle physics, and this new term changes the equations of GR too because the energy-momentum tensor of the energy of these new particles appears in the equations too.

So, you can have equations for GR + standard model without dark matter, and GR + standard model + some dark matter model. The equations and the solutions will be different.

This suggests nothing about GR. Except that one needs some theory about matter too, if one wants to have the complete equations, and, moreover, assumptions about its distribution, if one wants to make some nontrivial predictions. (Which is essentially a triviality. Say, Newtonian gravity does also not tell us anything about the Solar System without assumptions about the masses and locations of the Sun and the planets.)

I am wondering why it hasn't been rejected in this case, due to the missing mass issue.

Given that for the complete equations we need as a theory of gravity, as a theory about the masses (say, the SM), a failure of the resulting predictions cannot be simply attributed to one of them. Both may be the cause of the failure.

This is a quite general problem of experimental physics. In any real prediction, a lot of different theories are involved. Starting from the theories about what the particular measurement devices really measure, with which accuracy. If a single experiment fails, there are always many candidates. The way to solve this problem is to use many different experiments. So, say, measurement devices will many times measure what they are assumed to measure, compared with other measurement devices (of the same as well as of different construction), and other runs of the same to identify the measurement errors independently, so that their errors can (at least in principle) be excluded easily.