1) Does it use a standard headphone jack? That is, can I use SWMBO's noise-canceling Bose headphones with it sometimes?
Yes with some limitations. The volume controls are actually on the headphone lead and you need to get a special adapter which I'm not sure has been released yet.
2) Without a display, how do I navigate to a specific track?
Skip skip skip skip skip
3) Can it remember my playlist location on power-up the way a "regular" iPod does?
It should do.
5) Can an iPod Shuffle battery be replaced by mere mortals, or is that a specialist job?
You need to be able to solder to replace the battery. There are guides online that show you how to open it.