I can’t answer all of Will Baude’s questions, but I’ll give two of them a shot:
Why is a turnpike called a turnpike?
For that matter, what exactly makes a particular stretch of limited-access highway a turnpike?
Turnpikes were originally named “turnpikes” because that was the name of the turnstiles that were used at the toll gates; they started out as “turnpike roads” but the name was shortened to simply “turnpike” or even (particularly in the South) “pike.”
In general, a modern turnpike is a fully-controlled access highway (what engineers and Californians call a “freeway,” Britons would call a “motorway,” and francophones call an autoroute) that charges a toll for use; however, there are exceptions—most notably, the Connecticut Turnpike (part of Interstate 95), which stopped charging tolls after a nasty multivehicle accident at a tollbooth in 1985. Also, some contemporary turnpikes only charge tolls on part of their length—the Maryland Turnpike starts near Baltimore and runs to the Delaware border, but the toll is only charged at one location on the route.
So, in sum, the name “turnpike” is generally applied to roads that are, or used to be, toll roads, and there’s no particular logic to whether or not a particular toll road will be called a “turnpike.”