They are called paternosters because the cabins lined up and slowly moving remind of praying the rosary with the help of beads on a string.
It sounds like paternosters might have been rare (if not entirely non-existant) in the U.S.; I’d certainly never heard of or seen one here.
On the other hand, we do have the similar-but-even-scarier device here called a manlift, though they’re rarely (if ever) used in places where they’d be used by the general public (I think they’re only used in industrial applications). A manlift is basically the same idea as a paternoster, except that there aren’t elevator cars – there are just handholds and footrests, and to use one, you essentially step out over an open hole in the floor to grab onto it.
I rode on one, a few times, when I worked for Quaker Oats in the late 1990s; our big grain milling plant in Cedar Rapids, Iowa, was six or so stories tall, and had several manlifts. When we office-dwelling marketing people from Chicago would visit the plant, the manufacturing team enjoyed hazing us by making us ride on the manlift. I don’t like heights, but I managed to ride on the thing without falling off. 
Edit: Here’s a video on Youtube showing how they work: https://www.youtube.com/watch?v=AUdL_st3FFw
Bert Nobbins:
Basically, you use water that has been heated before the Sabbath and kept hot.
Many Jewish homes have electric hot water urns that keep the water hot and you’re simply dispensing it into your cup. Another option is an ordinary tea kettle filled with heated water kept close to a fire on the stovetop.
After dispensing the water from the urn/kettle into a cup, it then needs to be poured into another cup, and then (according to some) a third cup, before you’re allowed to put a tea bag into it, because that cools the water down enough so that it is not (by halachic definition) cooking the tea leaves (but it’s still hot enough for the flavor to be effective).
dzeiger:
Walk down the ten flights of stairs. Sure, it’s some effort, but at least you’re not fighting against gravity.
Except, of course, for the higher risk of accidents…
Many tea afficionados would argue that water at anything less than boiling (i.e., water that’s cooled down at all) does not, in fact, result in acceptable tea flavor. I suppose those afficianados, if Jewish, must content themselves with waiting until the Sabbath is over.
And what form of “work” would be violated by re-filling the urn, that wouldn’t be violated by pouring the tea into the cup? It’d be carrying, of course, but (presumably) entirely within one’s home.
Chronos:
Afraid so. On Jewish holidays, cooking is allowed because the essence of the holiday is “rejoicing.” On Sabbath, even though we are supposed to eat and drink festively, the essence is “resting” and the physical pleasures being enjoyed in its honor are secondary to that.
Re-filling the urn would not be allowed on Sabbath, because you’d be boiling the water. Pouring hot water into a cup is not cooking.
Other tea afficionados argue that you do not actually want boiling water (that is, 100C) but slightly less than boiling, and may even specify different temperatures to optimize different varieties of tea.
But either way, you have to heat water.
So, while we’re here asking this kind of question:
What do you do if you cross the international date line and miss Saturday
(Yes, I know it’s difficult to do, but it’s a common expression, and you’re waiting on one side, with two stars visible, and you – step – across…)
Do you take an extra Sabbath, out of respect, and to wind down from the week, and to recharge for the next week? Or do you just say ‘heh’. I’ll make it up with a really long Sabbath later?
I’m not sure it’s possible, logistically, to entirely miss Sabbath by doing so, but if you can manage it, I don’t think you’re obligated to keep Sabbath if it is never the Sabbath in the location you’re actually in.
However, you should note that this would not apply to the secularly-recognized International Date Line, but rather to the line that Jewish Law considers to be the Date Line. Where exactly that is is something of a question (there are two opinions on that), and in general, Jews find their best option is to avoid the question if at all possible. I myself cut a Hawaiian vacation short by a day in order to avoid that sticky question.
BTW, I finally located an archived copy of the Mailbag article I did for this site on the subject of the Sabbath:
That means that only one rabbi has weighed in on the question? 
The obvious options I can see are to use the same line as the secular one, to rotate the secular one by 90º to account for the Jewish day ending at sundown rather than at midnight like in secular convention, or to use the longitude 180º away from Jerusalem.
If you hired a man to march in the Pride parade for you on the Sabbath, would that make him a gay goy guy?
Oldie but goodie:
First Jewish astronaut comes back, they ask him how it went. Answers: "Oy…non-stop… Shachris, Minchah, Maariv…Shachris, Minchah, Maariv…Shachris, Minchah, Maariv…
[Shachris, Minchah, Maariv are the morning afternoon and evening prayer times.]
About the real decisions for real about this the rabbis have been there, done that.
I read about it regarding the Israeli Ilan Ramon, who died in the Columbia shuttle.
I’ll look around for more info. Another issue was geographical: Jews face Jerusalem when they pray, and you’ve got a problem there too.
In one of his Star Trek novelizations, Alan Dean Foster has the Muslim foreman of an orbital shipbuilding team pray towards Mecca when it passes below. Not every time, though.
Chronos:
Cute 
In all seriousness, there are more than two opinions (and Rabbis), but the two I was referring to are the major ones. Rather than going through all the various reasonings, I will just post a link to this document which discusses the different opinions (three, not just the two I had been referring to as the major ones) and it also includes discussions of such questions as crossing over the date line on the Sabbath or on other significant dates.
BTW, you are correct that one of those opinions is 180 degrees east/west of Jerusalem.
Bergen County, NJ still has these laws on the books - and it’s got a substantial Jewish population (my husband is from there, and still has family there). There are periodic attempts to repeal them but so far, nothing. Certain stores are exempt - grocery stores for example, and I gather you can purchase periodicals but not actual books (so as my MIL said: you can buy Playboy but not the Bible).
And it means that an observant Jew has very little time to shop.
I gave up trying to figure out the rules long ago. My wife is Seventh Day Adventist, and they just don’t get why we get into such legalistic detail. Just rest!
Now I get why they did what they did. It actually has to get legalistic when the rules have to be enforceable and agreed upon, which is how Jews lived for centuries. But I’m not playing that game. Sabbath is a day off from work of all kinds, and any pleasures I want to engage in that aren’t strenuous are just not “work” by any reasonable definition consistent with the spirit of the Sabbath.
I’m pretty sure I saw a manlift in lots of North American arcades in the 1980s. 
Re “paternoster” elevators – not sure whether I’ve encountered one at first-hand; but I happened to know the expression (and I agree - reckon it charming) from having, at school, read a short story by the German author Heinrich Boell, in the original German – in which a character regularly uses such an elevator (in Bavaria IIRC), and has a bit of a mental “thing” about it.
Luckily for me, the book containing the story – edited for those learning German – had an explanation in English, of the p-word in this context. I’m not Catholic, and half a century-odd ago, the word “paternoster” was meaningless to me until I chanced upon this reference.
adaher:
It’s more than just a matter of enforcement. It’s a love of G-d that (at least, ideally) leads Jews to examine his words for minute details of how to follow them. The fine detailing on the Mitzvos we perform are like fine detailing on a painting or a sculpture.