You don’t necessarily need a submarine and you certainly don’t need sonobuoys.
All you need is a vessel equipped with a fathometer, which is a pretty ubiquitous piece of equipment on seagoing vessels.
I would think thermoclines would render most surface echo-ranging pretty meaningless.
Here’s an idea: design a simple, disposable sonobuoy. Drop it in, untethered, into the ocean, have it ping like crazy on the way down, gathering all the information it can. When it hits bottom, it releases a floating transmitter that rises to the surface and broadcasts all the info. Drop a bunch of them in a grid formation, several miles apart, to cover the entire area being surveyed. Get the data, process it, and you’ll have an accurate map of the sea floor.
From my own (shallowish water) experience, we mitigate the impact of thermoclines and haloclines using vertical sound velocity profiles throughout the water column. The SVP correction is applied to the multibeam raw data. I think similar methods are used in oceanographic surveys.
When the Navy does collect their own mapping data, what do they do with it? Do they share it with international organizations, or do they carefully hoard it as a strategic secret?
The only place I know of where submarines do bath surveys is in the arctic under the ice. Surface ships are much cheaper. sonobuoys can’t measure water depth to the required accuracy nor can they hold anywhere near enough data to be useful. Autonomous underwater vehicles can and I believe do take bathymetry measurements, but they operate so slowly compared to surface ships that they are only useful in denied areas.
No, while thermoclines distort the signal path, it is easy to model and compensate for. And for deep water, the thermocline is such a small fraction of the total distance errors from that are insignificant.
It depends. 
as you can imagine. There are at least 3 kinds of surveys. Coastal or EEZ (exclusive economic zone) surveys which are done in cooperation with the host nation. What happens to the data is up to them. Not surprisingly, many nations see no reason to share the high resolution data. Open ocean surveys are eventually publicly available. Strategic surveys are never going to be made available. Those are taken in SSBN patrol areas and are very high resolution. Each of those factors alone make them highly classified.
The problem isn’t that a technological breakthrough is needed to allow the Navy to map the ocean floor. The problem is that the ocean floor, while mappable with current technology, is huge. Not only is it huge: it is also in parts of the world we do not own. So you can send someone/something to map a span of ocean floor, or you can send someone/something to an area of ocean without being detected. But you can’t reliably do both because the sort of active transmissions of sonar needed to accurately map the ocean floor are not compatible with stealth.
ETA: Just as an oh by the way, I’m not an oceanographer, I’m not a submariner. I didn’t even stay at a Holiday Inn last night. But I did spent 19 months on a minesweeper, and one of the things minesweepers do in peacetime is conduct surveys of the ocean floor for… reasons, sometimes using towed arrays with side scan sonar.
This is very cool.
Do minesweepers still accomplish that task by heaving a paravane over the side? If so I could see how that skill could come in handy in manhandling towed arrays.
That is the “sweep” part of it, yes (although we used cranes, not physical manpower to deploy the paravanes). But then actual minesweepers are becoming increasingly rare, at least in the US and UK navies, and it’s a point of contention whether it’s worth it to build ships with such equipment. What has been getting built lately is more in the realm of “minehunting,” which comes without the sweep gear but instead relies on sonar and such to spot mines in the water and then deal with them one way or another (just not by “sweeping” them, because they don’t have that capability).