Let me add a few comments that might fill in some gaps not covered by other answers.
- When would it be safe to say that gravity has been successfully quantized?
A quantum theory consists of an algebra of "observables" (taken often to be a C*-algebra), a "state"---positive linear functional on the algebra, and a way of assigning a Hermitian element of the algebra to each conceivable single-number-outcome experiment. The theoretical prediction for an experiment is given by evaluating the state on the corresponding observable---taking its "expectation value". Higher moments of the same observable give information on the statistical distribution of experimental outcomes under identical conditions. Of course, since physicists don't have perfect information about the state of the entire universe, we are forced to work with a class of states and test hypotheses about which are closest to fitting all available observations.
If the above construction can be carried out such that all experiments sensitive to gravitational effects have representative observables and all their expectation values with respect to states that are approximately classical reproduce the results of classical general relativity, then the result is a successful quantization of gravity.
- Has the above construction succeeded or failed for gravity any more or less than for any other field theory?
This construction has been mathematically rigorously successful for free fields on Minkowski spacetime (and one some classes of curved Lorentzian manifolds, aka spacetimes). Other examples of rigorous constructions exist, but they are not of direct physical relevance. The golden standard of physically relevant field theories is quantum electrodynamics (QED). It has not yet been rigorously constructed. However, assuming that QED exists as a quantum theory and that all expectation values of observables in it can be expanded as power series in the strength of the interaction (electron charge), then each coefficient can be constructed order by order. Existence of a family of quantum theories corresponding to such a series expansion has not been resolved at the mathematical level.
The above construction is referred to as "perturbation theory". Essentially, it's a procedure that starts from a rigorously constructed free theory and constructs an interacting theory, in the above sense, as a power series in a finite number of parameters (coupling constants, which measure the strength of the interaction). This transition from free to interacting (for a given set of coupling constants) is not necessarily unique. Theories for which this non-uniqueness is parametrized by a finite number of parameters are called renormalizable, the rest are called non-renormalizable.
QED and the rest of the standard model have been found to be renormalizable, but perturbative gravity, obtained by expanding around Minkowski spacetime, has been found to be non-renormalizable. In principle, the non-uniqueness in the perturbative construction of interacting field theories needs to be fixed from experimental input. The unbounded number of experimental inputs needed to construct a particular realization of a non-renormalizable field theory is what makes such theories unattractive.
An even bigger failure on the part of gravity is that the classical structure of general relativity (a Lorenzian manifold) enters in an essential way into the construction of free quantum field theories. If it is omitted, we have no way of guaranteeing the correct classical limit of the resulting quantum theory. On the other hand, a Lorentzian manifold should be the result of a classical limit of quantized gravity, not an input to it. This problem has not yet been resolved at the level of theoretical physics.
- What role do Euclidean functional/path integrals play?
The answer to this question is that it is unclear. Euclidean path integrals are equivalent to other constructions of quantum field theories if the space-time manifold has translational symmetry along the time direction. In such cases, given a Euclidean path integral formulation of a field theory (as mentioned in A.J. Tolland's reply), the corresponding quantum theory can be constructed and vice versa.
However, in the case of the gravitational path integral, if it is used as a starting point for a non-perturbative definition, it may not correspond to a quantum theory at all. Since the functional integration is performed over all possible Lorentzian metrics, there is no background time-translation symmetry to speak of. Therefore, even the construction of the Euclidean path integral succeeds, the equivalence to a quantum theory may fail. Again, this question has not yet been resolved even at the level of theoretical physics.
- Are discretization or other non-trivial modifications of the underlying manifold structure of spacetime necessary?
Unknown. However, there is no direct evidence that gravity cannot be formulated as just another field theory on a smooth manifold. In fact, there are counter examples: the infinitely many possibilities for perturbative quantization of gravity on Minkowski spacetime. Recall that non-renormalizability does not mean non-existence, just non-uniqueness. The many proposals that try to do away with a smooth manifold are heuristic at best.
If anyone is interested, I can elaborate on any of the points I've brought up above.
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