Does gravity sometimes get transmitted faster than the speed of light?
No, gravitational influences never travel faster than the speed of light. However, a naive incorporation of a speed-of-gravity delay would actually lead to the Earth's orbital motion speeding up, not slowing down. (Think about the geometry carefully.) I explained here why that doesn't actually happen in general relativity.
Cuckoo asked: So it appears that the force of gravity is indeed directed towards the current orbital position of Earth, without accounting for the delay caused by the speed of light. How is this possible?
If the motion is straight or circular the aberration cancels out, see Steve Carlip: Aberration and the Speed of Gravity:
Steven Carlip wrote: The observed absence of gravitational aberration requires that "Newtonian'' gravity propagates at a speed ς>2×10¹⁰c. By evaluating the gravitational effect of an accelerating mass, I show that aberration in general relativity is almost exactly canceled by velocity-dependent interactions, permitting ς=c. This cancellation is dictated by conservation laws and the quadrupole nature of gravitational radiation.
or to quote the Wikipedia article on the subject:
Wikipedia wrote: Two gravitoelectrically interacting particle ensembles, e.g., two planets or stars moving at constant velocity with respect to each other, each feel a force toward the instantaneous position of the other body without a speed-of-light delay because Lorentz invariance demands that what a moving body in a static field sees and what a moving body that emits that field sees be symmetrical. In other words, since the gravitoelectric field is, by definition, static and continuous, it does not propagate.
Let's temporarily pretend that we can speak of gravitation as being released in pulses, which is a weird way to speak, but I think is part of your mental model. Let's also pick a reference frame : suppose the Sun is stationary in our laboratory.
The Earth right now is not reacting to the right now pulse of the Sun's gravity. The Earth right now is intercepting the pulse of gravity released $8$ minutes ago, which happens to point directly back to the Sun (because it is stationary).
For a careful, mathematical treatment of this, see The Feynman Lectures in Physics, vol. II, section 26-2, where an electric field in laboratory coordinates is found to have magnetic components in the moving frame which (to first order) cancel the aberration (angular deflection) caused by retarding waves by their travel times. The same thing happens in gravitation: the off-diagonal elements of the tensor pick up the terms necessary to cancel retarded aberration (to first order).