FindShortestTour and the algorithms therewithin implemented internally?
Also: Can I access some form of incremental, in-progress result during evaluation? (e.g.: PCB autorouting)
There are many unrelated methods that
There's more information for some of these under the Options section.
I'd suggest to Google a bit more for the methods that you are interested in to learn how they work, and asking about one specific method. Some of these methods are completely unrelated.
The reason why so many methods exist is that no fast (polynomial time) method is known that is guaranteed to give the shortest tour. For this reason there was a lot of research on coming up with good approximations to the solutions: finding "short enough" tours, if not the shortest one.
If you need the specific details of one of the methods listed here, WRI support is usually pretty good in supplying the appropriate references.
Unfortunately I don't believe you can access a partial solution and continue the computation from there.
Here are my guesses on what the various methods do: