The semi-classical Einstein equations (without a cosmological
constant) are $G^{\mu \nu} = 8\pi \langle T^{\mu \nu} \rangle$. I am
told that there are serious objections as to why these equations
cannot be a completely correct description of nature. (If they are
correct, then there is no need for quantum gravity.) From what I have
heard, the objections are [...]

There are many, many issues with semiclassical gravity. Examples of problems in addition to the two you mention: questions about stability of flat spacetime, problems with the Born rule and Copenhagen interpretation due to nonlinearity. The issue is not whether semiclassical gravity is fundamental but whether it's even a reliable approximation, and, if so, which formulation of semiclassical gravity to use and how to put bounds on the errors.

To my mind, there is an overriding physical reason why semiclassical gravity clearly can't be a fundamentally correct theory, which is that it couples a classical field to a quantized field. Physicists realized almost a century ago that this just generically doesn't work. Bohr and his followers wanted to keep the electromagnetic field classical while quantizing the atom. This resulted in the Bohr-Kramers-Slater theory, which lived for only a few years before being disproved experimentally by Bothe. The basic issue is that if you want energy and momentum to be conserved in a quantum context, you need long-range correlations, and those correlations don't exist in a classical field. For example, in the photoelectric effect, you would like the absorbed photon to give its energy to exactly one electron. If it gives its energy to both electron A and electron B, then you've doubled the amount of energy that's present. What prevents this from happening is that the electromagnetic field is quantized, and therefore it can have correlations between what it does at A and what it does at B. Without these quantum correlations, energy and momentum can at best be conserved on a statistical basis. Nonconservation of energy-momentum has never been observed, and if it did happen (locally), it would render the Einstein field equations inconsistent.

Do the Hawking-Penrose singularity theorems still hold in semi-classical gravity ?

Willie Wong has answered this, but I would like to provide some perspective in addition. Semiclassical gravity, if it works at all as an approximation scheme, is an approximation scheme that applies to weak gravitational fields. Therefore if it told us something about singularities, we wouldn't believe it.

Generically, theories of quantum gravity tend to provide mechanisms that prevent the formation of singularities once you get down to the Planck scale.

So what if superluminal propagation is possible in extreme situations ? What evidence do we have against it anyway (in such extreme cases that is)

I think Willie Wong's answer clarifies why this is not really the right way to frame the issue. However, there is a generic objection to superluminal propagation, which is that if it happens, it typically means we don't have existence and uniqueness of solutions to Cauchy problems, so that physics loses its predictive value.