I don't know why you are going to such inordinate lengths to deny the obvious but that's your affair.
The facts are these and they are verifiable (based on unclassified public domain knowledge):
1. From various accounts, inertial navigation systems on modern submarines are phenomenally accurate and have been so since at least the 1970s. The absolute accuracy depends on (a) the initial calibration and the location of that calibration, (b) on the path or trajectory of the submarine (cumulative errors inherent in the inertial navigation system and charting errors, etc.) accumulated over the tour of duty; (c) unique/specific calibrations for certain operational requirements, locations etc. All this assumes the sub remains submerged and has no external calibration reference (but the fact that it remains submerged doesn't necessarily mean that it doesn't have the means to re-calibrate its inertial navigation—as it often does!).
2. The length of time a submarine can remain submerged and operate safely† on only its inertial navigation systems sans external calibration is usually classified but indications are that it can be months for nuclear submarines (again, much depends on operational requirements and the locations of those operations, the technical parameters of the inertial navigation, etc.).
† As discussed—excluding uncharted seamounts, etc. (whilst a rare event, we've now seen it happen twice within 16 years—the USS San Francisco being the first to collide with one).
3. However, the basic inertial navigation system is an open-loop system and thus subject to both long-term drift and long-term calibration imprecision—and other 'compensation' calibration stuff can only go so far in holding accuracy. Eventually, a submarine's inertial navigation system needs to re-calibrated. That said, in recent years, reports indicate newer inertial navigation technologies have led to even further substantial increases in open-loop accuracy. Obviously, the precision of this accuracy is classified.
4. Since the 1960s nuclear submarines have been able to communicate with the outside world whilst remaining submerged using VLF radio, https://en.wikipedia.org/wiki/Communication_with_submarines. Years ago, I used to take an interest in this and I could detect the RF carriers from the high power transmitters (≈1MW) used to communicate with submarines on ordinary radio equipment but obviously the modulated data they contained was meaningless to me. I've now only a brief public domain knowledge of how these days, say, the US uses this and associated newer communication technologies for calibration but you don't have to too bright to know that using atomic clocks you can triangulate a position with considerable accuracy (as per GPS). Using geopositioning in conjunction with gravity gradiometry, etc. cannot only act as a backup but it can also improve the 'local' location picture/accuracy.
5. "Under #4, the first one is about navigation. Again, if you don't know somethings't there, then this doesn't help you. The second one is about detecting other submarines. This was an underwater mountain."
I cannot understand why you cannot picture this. If you can detect another submarine by its small, somewhat reduced local gravity signature then you can also easily detect an unknown or uncharted seamount by its hugely increased local gravity signature (rock being denser than seawater and it being much larger than any enemy submarine is ever likely to be (ipso facto, as the sub's gravimetric detection equipment is capable of detecting a comparatively small submarine then it must also have a detection sensitivity many times more than is necessary to detect a huge seamount). Moreover, a submariner can use the very specific signature of a known seamount to re-calibrate his sub's inertial navigation system—as no two seamounts would ever have exactly the same gravimetric signature (and if by some billions-to-one chance they ever did, then they wouldn't be anywhere near each other to cause confusion).
For example, say a submarine has not surfaced for months and its captain has orders to fire a missile on date xyz. Whilst he knows pretty much where he is from the sub's inertial navigation system, its drift over the several months the sub's been at sea is not precise enough for him to determine the precise location needed to launch the missile (and the inertial navigation system of the missile also has to be calibrated/to be precise). He cannot surface for fear of being detected so what does he do to re-calibrate? Simple, he heads for a seamount (or some other underwater phenomenon) whose location and gravimetric signature has already been properly charted during some earlier survey, and he uses the sub's inertial navigations system to get him into the near vicinity of said seamount—as it's still accurate enough to do that with ease. Moreover, the sub won't collide with the seamount as its precise location is now clearly visible on the sub's gravimetric equipment. As he already knows the EXACT location of the seamount, he now has the perfect means to recalibrate the inertial navigation system. At no time has the sub surfaced, nor has it broadcast its position, as both of its underwater navigation systems are completely passive.
6. The links I provided were a first-pass effort, and sure, it's dead easy to nitpick them as you've done. The fact remains that all the references I've provided are relevant in that they all refer to various technologies that are integral to both submarine geopositioning and undersea collision avoidance. Separating them out is about as stupid as saying 'I'm going to build a computer but with only CPUs—RAM and monitors are off the menu, I won't be using them'.
7. "All that lets us know it isn’t ‘child play’." I beg to differ, when a technology has been in use for going on 50 years and has been through multiple stages of evolution of over this time to the extent that now commercial operators are using it on a day-to-day basis on their undersea ROVs etc., then essentially it is child's play to the big players such as DARPA, the Military and so on (however, that's not to say, there aren't any remaining difficulties). What we currently know from public domain sources is that this technology is already very sophisticated and it's not just theoretically possible, as it's working in practice and it has done so for decades. One would have to very stupid not assume that it's now much more so (given it's highly classified nature). Did you actually take the time to read this: https://en.wikipedia.org/wiki/Gravity_gradiometry ?
"Gravity gradiometers
Lockheed Martin gravity gradiometers
During the 1970s, as an executive in the US Dept. of Defense, John Brett initiated the development of the gravity gradiometer to support the Trident 2 system. A committee was commissioned to seek commercial applications for the Full Tensor Gradient (FTG) system that was developed by Bell Aerospace (later acquired by Lockheed Martin) and was being deployed on US Navy Ohio-class Trident submarines designed to aid covert navigation. As the Cold War came to a close, the US Navy released the classified technology and opened the door for full commercialization of the technology. The existence of the gravity gradiometer was famously exposed in the film The Hunt for Red October released in 1990."
8. I'm not responsible for why so many are still in WWII mode and are only thinking of [the limitations of] active sonar. Perhaps the Military has been smarter than I've given it credit for, in that it seems to have managed to keep an understanding of this technology even out of the eyes of many techies.