English is tricky here. What is ŝtuparo in Esperanto, I normally call in English "the stairs." Stairway and and even more staircase give me the idea of the room where the stairs are in—including the landings. I would call that in Esperanto "ŝtuparejo." Perhaps a native speaker of the English can tell more about it.
Now there's an interesting linguistic oddity; stairway struck me as a word I barely associate with "staircase" at all. Looking it up, I find that while the OED makes it synonymous with staircase, Webster's Collegiate (a very American dictionary) says "way up and down a staircase;" and I believe here in New England one only sees it in the context of "don't block the stairway."
That is interesting - thinking about it, I wouldn't see 'staircase' and 'stairway' as synonymous. Like you say, someone can "block the stairway" but to me that includes the whole area that the stairs are located in - the equivalent to a passageway except that it goes up and down and has stairs in it. The staircase I think of as the physical collection of stairs. You can have a spiral staircase, but would you have a spiral stairway? That doesn't sound right.
As a midwestern American speaker, I would define staircase as the elaborate stairs found in a mansion, or cruise ship. And I would define stairway as the functional stairs found in a college or office building. It is like the difference between a bookcase and a bookshelf.
As a New Englander, I'd call an elaborate one a "grand staircase." I'd call anything a staircase that has its own well and has more than one flight - has a landing and / or a turn. Anything less is stairs (which is also the general term) except for a set of just a few, or a set that are outside; those are steps. (I suspect New England housing stock tends to be more vertical than other parts of the country.)
For me, the stairs are either a collection of individual stair "units," or one whole flight--E.G. "The book is still sitting on the stairs." This is what I think of when I think ŝtuparo. ŝtuparoj would be multiple flights of stairs, and ŝtupoj would be many stair-units, E.G. "He had only five stairs until the next landing." I've always thought of a staircase as the two side supports that run from the top of the flight to the bottom, between which each stair runs. A stairway is simply the area in front of the stairs--the way to get to the stairs, whereas a stairwell is the vertical shaft in a building containing all of the flights of stairs and their connecting landings. I usually think of this as an enclosed space with many levels, where one can look between two flights and see from the top to the bottom, or vice versa. I would not consider the entryway to a raised ranch, where one can only either go down to the basement, or up to the living room/kitchen a stairwell. I am from Chicago, but this doesn't mean this is how most Chicagoans see it.
The better translation of ŝtupo is step. So a line or group of steps would be a ŝtuparo. The main point being this -- if someone said, "Metu la libron sur la ŝtuparon, mi petas." would any functioning Esperantist who also speaks English wonder whether he or she ought to be looking for stairs, a staircase, or a stairway? More than likely this person would go over to where there was a group of steps and put the book there.