1 - Can a quantum field, or anything really, have absolute zero energy in any local point?
It's well known that zero-point energy is not actually zero. The ground system of a quantised classical system has non-zero energy.
However it turns out with super-symmetry, we can actually obtain such a ground state. Unfortunately, so far the only evidence for super-symmetry has been theoretical, and not experimental...
2 - What is the "nothing" the Universe came from in the terminology of the advocates of the zero-energy universe theory?
Typically advocates of a universe that came into existence from nothing by a quantum fluctuation fall on this very point. It's a hugely speculative idea, and for that reason, should be taken with the degree of regard reserved for hugely speculative ideas, ie not very much.
3 - Why is it correct to attribute a negative value to the energy of gravity, would it not be possible to say that the laws of conservation of energy are wrong instead?
Its actually quite difficult to come up with a generally covariant form of the conservation of energy; and more-over, it hasn't been universally held; for example, Fred Hoyles and Hermann Bondis Steady State Theory, which was the leading cosmological contender before the Big Bang held that there was a cosmological creation of energy. Caroll writes:
The point is pretty simple: back when you thought energy was conserved, there was a reason why you thought that, namely time-translation invariance... [however], in general relativity that’s simply no longer true. Einstein tells us that space and time are dynamical, and in particular that they can evolve with time. When the space through which particles move is changing, the total energy of those particles is not conserved...and one that has been experimentally verified! The success of Big Bang Nucleosynthesis depends on the fact that we understand how fast the universe was expanding in the first three minutes, which in turn depends on how fast the energy density is changing. And that energy density is almost all radiation, so the fact that energy is not conserved in an expanding universe is absolutely central to getting the predictions of primordial nucleosynthesis correct.