I'm approaching this by way of applying some basic science to the thing: since it is a fantasy world, science doesn't have to apply, of course.
Being tidally locked does not make the sun brighter on the day side. It is merely more persistent. The brightness in the center of the hot side is still only what it would be at Noon if the planet were spinning.
Also, Gliese 581, our example, is not like our sun. It is a red dwarf, far smaller, dimmer, and cooler than our star. The back-of-the-envelope estimates I've seen suggest that, if the planet had an Earth-like atmosphere, you might see high temps of around 160 degrees Fahrenheit on the day side, and -25 on the night side.
The habitable band around the terminator between night and day, then, is probably pretty wide, as even the hottest and coldest points aren't all that hot or cold. There are no fire storms - 160 degrees isn't hot enough to ignite most materials. Water tends to evaporate quickly, but it doesn't outright boil. The cold.. well, -25 is something that folks in the Northern US see not too infrequently.
Another thing you'll see: persistent wind. On the day side air is heated, and will rise. Cooler air from the night side will be sucked in to replace it. So, near the ground, you get winds flowing dark to light, and up high you get flow from light to dark. This convective mixing may mitigate the temperature difference significantly. Storm patters would not be much like what we see on Earth.