Or the Superman story, “Let My People Live?”
If the night pole is cold enough, air would freeze onto it until the atmosphere was gone, condensed into a big frost glacier. The only thing that would keep the pole from getting that cold is convection - air carrying heat from dayside to nightside.
Therefore there must be a critical density for the atmosphere to exist. Below that and it will not be able to transfer heat fast enough and it will all freeze out. I don’t have a sense of what that critical density would; it might be above what we have here on Earth.
I remember a Traveller 2300 module about a moon that was orbiting a gas giant / brown dwarf and tidally locked. Was it called Aurora?
And if the hot side is hot enough, a significant number of molecules will acquire enough kinetic energy to escape from the planet entirely. So having an atmosphere on a tidally locked planet might be a bit unlikely (which is what I was getting at in post #2.
A plot point in one of Niven’s Draco’s Tavern stories; the chirpsithra mentioned that during an ancient war they deduced their enemy came from a non-tide locked planet because their prisoners of war hated the chirpsithra’s idea of proper biosphere simulation - which included constant extremely high wind.
More than that: The discovery of its rotation came in between the story’s acceptance and its publication. Niven frantically called up the publisher telling him to cancel it, because it had been outdated. The publisher said, who cares, it’s still a good story.
Thanks for all the answers !
So there’s even a word to speak of the concept. Durn scientists and writers stealing my original ideas ! 
I’ll admit, I hadn’t thought about the weather and climate at all (beyond the scorching/freezing thing, anyway) since they weren’t modelled by the game, but it’s an intriguing question. I would indeed expect there to be, if not constant 120 mph winds, at least lots of turbulence and storms in the habitable zone since it would pretty much be a permanent convection hotspot provided there is an atmosphere to begin with, wouldn’t it ? Hot air trying to move “nightside”, cold air trying to go “starside”, always colliding in the middle even if the air currents mostly aren’t at the same altitude, that sort of thing ?
How about clouds ? Would clouds form at all on a tidally locked planet ? I’d expect the starside to be well out of any trace of water relatively quickly in the planet’s lifecycle - straight up boiling into the atmosphere, then either raining on the equator or snowing nightside, but I’m no good at this climate science stuff 
I daydream about life on the habitable ring… the people there weren’t natives, they were all colonists from other worlds and some of them were bog standard humans from good ol’ Earth. We crave the familiar. I figure that there might be a band on the edge of starside where it’s both habitable and more or less “daylight” rather than twilight, and a symmetrical “night time” spot on the opposite side, so maybe people would just lead normal Earthling lives except they’d commute starside in the “morning” and have their homes on “nightside” ? Sort of turning distance into time, if you follow my meaning. But I’m not sure exactly how wide the equator might be, a 500 miles commute would probably not be practical, even in the far future IN SPACE !
Zelazny’s “Jack of Shadows” is another story set in a world (earth!) where it is similarly locked, but in the realm of magic/fantasy.
The planet Medea in Niven’s story Flare Time was also tidally locked to its sun
That’s not the only thing that makes it outdated. Inspired to find it by this thread, I read it this morning (it’s very short). The main character has a pipe attachment in his space suit so he can have a smoke!
Yes, upon further consideration I withdraw my earlier comment. A volume of “hot gas” separated from a volume of “cold gas” is in equilibrium, but it’s an unstable equilibrium; a lower-energy configuration would be “hot gas” on top, “cold gas” on bottom. You’d probably end up with a large-scale convection cell over the habitable strip, with the prevailing winds blowing from the cold zone to the hot zone.
And because of that, you’d have a nearly permanent wet and cold coldfront full of storms along the whole strip.
You can call the planet ‘Seattle.’
I’d expect a constant convection cell at the terminator. Air on the hot side expands and rises, flows at high altitudes over to the cold side, where it cools and condenses and then flows back to the hot side. You’d have hurricane-force wind blowing from cold to hot across the terminator. Coriolis effects might cause it to twist into hurricane-like storms here and there, which might even be semi-permanent features. Mountain ranges will also locally block or divert some of the wind, which might also cause local permanent storm systems to form.
The main challenge will be that the cold side of the planet will act as a permanent water trap. Water which gets blown far enough to be away from the storm zone at the terminator will fall as ice or snow onto the cold side, and then stay there forever. Eventually all of the water on the planet will be frozen on the cold size, and the hot side will be a permanent desert. If you have oceans over most of the surface, and maybe some geothermal heating of the bottom of the ocean on the cold side, that can cause enough movement of the water to maintain some sort of hydrological cycle. In which case I’d expect constant standing storms across the terminator, and maybe massive glaciers slowly flowing from the cold side to the hot side in places.
The pat answer I recall from discussions about this was that the dark side would be cold enough to make the atmosphere condense; so eventually the atmosphere would end up all pooled / piled up on the cold side. Of course, it depends on the convection current and maybe topography. One imagines rivers of liquid nitrogen and oxygen flowing sunward until they evaporate into atmosphere again in warm air currents, etc.
A tidal-locked planet must have its axis perpendicular to its orbit. The habitable band is a circle connecting the Poles.
And so the Sun would be over the western horizon for one hemisphere and over the eastern horizon for the other hemisphere.
Hence why the OP put the word “equator” in scare quotes. While it’s true that the axis of rotation would be perpendicular to the orbital plane, the inhabitants of such a planet would have no reason to use the axis of rotation as the axis of their coordinate system, and would almost certainly set up a coordinate system where the “equator” ran along the center of the habitable band.
Probably not - very low rate of rotation means very low Coriolis force.