Is Earth's orbit around the Sun affected by the ~8 minutes light delay?
You seem to have misunderstood some things about the work of Kenneth Nordtvedt (spelled that way, not Nordvedt). He is mainly known for pointing out that in some well-motivated alternative theories of gravity (i.e., not general relativity), the equivalence principle could be violated. Massive, self-gravitating bodies would have slightly anomalous ratios of inertial to gravitational mass. This is known as the Nordtvedt effect. The effect was searched for in lunar laser ranging experiments. The observations did not detect any Nordtvedt effect, and they placed an upper bound on it. This is discussed in section 3.6.1 of the review article Will 2006.
In any relativistic theory, an attraction or repulsion from an object at a distance $r$ is not toward the object's current location but toward the position of the object that would have been extrapolated from its state of motion at a time $r/c$ in the past. This is thoroughly verified by experiment and is not controversial. There is a discussion of this in Feynman, section II-26-1. In the sun's frame of reference, this extrapolation has no effect. In any other frame, the earth is seen to accelerate toward the position where the sun would have been extrapolated to be based on its position and motion 8 minutes ago.
Nordtvedt did not claim that gravitational interactions propagate instantaneously. General relativity's prediction that gravitational effects propagate at c has been accurately, although indirectly, verified by observations of the Hulse-Taylor binary pulsar. Solar-system tests are not capable of unambiguously testing this feature of GR; see Samuel 2003 and Will 2003.
It sounds like you've misinterpreted someone's explanation of the extrapolation idea. An equivalent way of stating the extrapolation idea is that if A's motion is inertial, then the force of A on B acts along the line defined by A's current instantaneous position, not its position retarded by $r/c$. This is not the same as saying that the interaction actually propagates instantaneously. None of this is a special, unorthodox theory proposed by Nordtvedt. Carlip 2011 gives a nice explanation. In the electromagnetic case, consistency is achieved because there is both an electric force and a magnetic one. In general relativity, you can talk about a similar type of gravitomagnetic force, and it plays a similar role in the analogous argument. Again, none of this is unorthodox or due to Nordtvedt.
Will 2006 - "The Confrontation between General Relativity and Experiment," http://www.livingreviews.org/lrr-2006-3
Feynman, The Feynman Lectures
Samuel 2003 - http://arxiv.org/abs/astro-ph/0304006
Will 2003 - http://arxiv.org/abs/astro-ph/0301145
Carlip 2011 - "Does Gravity Travel at the Speed of Light?," http://math.ucr.edu/home/baez/physics/Relativity/GR/grav_speed.html
There is no changing gravitational field. The solar gravitational potential through which the Earth moves has been set up a long time ago and continues to be present as the Earth moves through it. Consequently, there is no dependence on the time lag--this is just the same as why you don't need to worry about retarded potentials when deriving the energy levels of the hydrogen atom--the Coulomb field is static, and has been present for a long time.
And graviteomagnetic effects are still the same, too. They are due to an existent potential, and the Earth doesn't need to communicate with the sun to notice this potential.
The picture would be different if we were talking about, say, detecting whether the sun had been knocked out of it's orbit by a passing star. Then, it would take eight minutes for this change to be encoded in the local gravitational field.