Gravitons and general relativity

We don't have a quantum theory of gravity yet, so the honest answer to your questions is "nobody knows". But I think most people would guess that the quantum theory of gravity would bear a similar relationship to curved spacetime as the quantum theory of electrodynamics (QED) bears to Maxwell's picture of electric and magnetic fields. That is, spacetime would "appear" to be curved at large scales, but the underlying mathematical picture would be very different -- just as the mathematical description of quantum electrodynamics is very different from Maxwell's equations. On the other hand working engineers use Maxwell's equations every day, as they're a very good approximation; similarly, Einstein's field equations are going to be a very good approximation of whatever final theory of gravity there is.

Gravitational waves are probably explained as a collection of gravitons, just as electromagnetic waves are explained in QED as a collection of photons.

Black holes would still be described as "a region of spacetime which light cannot escape from", but the quantum explanation would be in terms of the strength of exchange of virtual gravitons rather than the extreme curvature of spacetime.

Gravitons are theorised by looking at the linearisation of a perturbation of curved spacetime and it turns out that it is a massless spin-2 particle. Hence as one commenter has pointed out, it presupposes curved spacetime.

However, having derived the graviton in curved space, we can consider it in flat space. According to Feynmans Lectures on Gravitation, if we also linearly couple the graviton to the energy-momentum tensor the full set of Einsteins field equations can be derived. Hence, the underlying space that we took to be flat, is in fact curved. Thos is entirely unexpected. In fact, Feynman writes in section 8.4:

The fact is, that a spin-2 field has this geometrical interpretation; this is not something readily explainable - it is just marvellous.

This argument did not originate with Feynman. As the introduction to the book makes clear, it was first considered by Suraj N Gupta in a paper published in 1954, and also treated by Robert Kraichnan in an unpublished batchelor's thesis at MIT. However, Feynman's derivation was independent of this earlier work.

A different argument was given by Weinberg in the mid-60s,who assuming 'reasonable analycity' properties of graviton-graviton scatterings, showed that the graviton could only be Lorentz invariant if and only if it couples to matter and itself universally, that is the strong equivalence principle is satisfied. From there, the rest of GR can be derived.

These are all fine questions, but it is important to get first a proper intuition.

Gravitons are to gravity what photons are to electromagnetic field, or what phonons (quasi-particles of sound) are to deformation. They are all excitations of some field (quantum field, to be precise). "Excitation" implies there is some default state of the field, which we imagine to be somehow "perturbed". We call the default state "vacuum" and perturbations - "particles" (because when there are few of them, they behave like regular particles). What constitutes a "vacuum" and what a "particle" is quite subjective, and some choices are better than others. We try to pick vacuum as a stationarry and most "calm" state of the field (uncertainty principle makes it impossible for the field to be completely calm). Classical stationary solutions to Einstein equations is a good "seed" for the vacuum of gravitational field.

Given the intuition, we imagine gravitons being superimposed onto the background of some quasi-stationary curved space-time. When gravitons act synchronously, we call it gravitational waves (in the same way as we call synchronized movement of photons as electromagnetic waves). Gravitons do not explain black holes, they are just quants of perturbations on the curved background.