This is not correct as you state it. Time dilation is not the same as spacetime curvature, nor is there a single direct relationship between the two.
You would do better to stick to the very simple statement you make next:
This is a reasonable statement (if we neglect the time dilation due to the Earth itself, and for that matter due to the Sun and the Milky Way galaxy and any other gravity wells that we are in but the black hole is not in--but those are minor corrections). But notice that there is nothing in there about "space time fabric", nor does spacetime curvature appear anywhere in the formula.
Also note that, as
@Ibix said, this formula is only valid for one very specific type of observer, an observer who is "hovering" at rest at a fixed altitude above the hole.
Yes, that is correct.
To make this comparison you need to do one of two things:
(1) Have the observer travel down close to the hole, hover there for a while, and then come back up and return to the Earth, and compare the elapsed time on his clock with the elapsed time on Earth clocks.
(2) Have the observer hovering close to the hole exchange round-trip light signals with Earth, and compare the round trip elapsed time on his clock with the round trip elapsed time on Earth clocks.