Eclipses
Many fans of Halo hold the belief that it will be frequently eclipsed by either Threshold or Basis, and that this will be a major factor in when Halo experiences night; however, I believe that this is unlikely. Threshold appears to cover an area of the sky spanning somewhat less than pi/6 radians when viewed from the ground, and Basis covers an area no larger than our own moon does. Accordingly, the region of sky through which these bodies move (from a coordinate system w. origin at the Halo but fixed with respect to the Basis/Threshold orbital plane) can be considered as a ring-shaped strip pi/6 radians across with the Halo at its center, 1/12 of which is occluded by Threshold and Basis at any given time. Clearly, when the sun lies in this strip of sky, it will spend at most 1/12 of its time eclipsed; the rest of the time it will never be eclipsed.
The question then becomes: how often does the sun enter this strip? Well, its distance from the strip depends upon the angle between it and the Basis/Threshold orbital plane. As an approximation, this angle varies over time as phi sin (2 pi t/T), where phi is the maximum angle between the B/T plane and the sun, and T is the length of the Halo's year. During a single revolution, this angle is between pi/12 and -pi/12 (that is, the sun is within the strip) when sin (2 pi t/T) is between pi/(12 phi) and -pi/(12 phi). So the fraction of a given orbit during which the sun is in the relevant strip is roughly equal to 2(arcsin(pi/(12 phi)))/pi, or 1, whichever is smaller. For phi equal to or less than pi/12 (15 degrees), the fraction is one; for phi equal to pi/6 (30 degrees), the fraction is 1/3; for the maximum of phi equal to pi/2 (90 degrees), the fraction is .1066. Battle of Hastings! Sorry.
As said before, when the sun lies in the strip it will spend at most 1/12 of its time eclipsed. However, half of that time you won't notice from the Halo, because it'll be night anyway. (Although it'll make the nights slightly darker, since the opposite side of the ring won't be glowing overhead.) So, the punchline: If phi is less than 15 degrees, then eclipses will turn the day into night AT MOST 1/24 of the time. If phi is greater, this fraction will be even smaller; for phi equal to 90 degrees, eclipses will occur during the day less than 1/225 of the time.
Put simply: Eclipses are common, but not that common. They never darken the day more than 1/24 of the time, and during certain times of year this may drop to 1/225 of the time.
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Seasons
So can we know anything about the Halo's axial angle, and therefore about the nature of the diurnal cycle? Well, if we ignore the effects of precession, we can treat the Halo as having a fixed axis of rotation, as the Earth does (over the short term). The angle then varies as the Halo orbits Soell, from 90 degrees to some minimum and back again, twice in each year. (Similarly, the Earth's angle goes from 90 degrees, at the equinoxes, to a minimum, during summer and winter.) The angle, in turn, affects the amount of heat and light the Halo receives from Soell. It does this in two ways. First, in the way already described, by varying the day length; when the angle is larger, the days are shorter. Second, by varying the angle at which sunlight hits the Halo and therefore varying the intensity of that sunlight. When the axial angle is smaller, sunlight hits more at an angle and its intensity decreases, just as it does at the Earth's poles (which is of course why they're so cold). When you combine the two effects, you find that the Halo receives the maximum amount of energy for an axial angle of about 75 degrees. At smaller angles than this, the days are longer but the sunlight is weaker, and so the Halo is in general colder; at larger angles, the sunlight is stronger but the days are shorter, and again the climate is colder. These effects concern the whole Halo; whereas the Earth experiences summer in one hemisphere at the same time as winter in the other; the entire Halo warms or cools at the same time.
Now, we can see from the screenshots that the sun approaches the 75-degree maximum, but we don't know if it crosses it. Thus the pattern of a Halo year can't be determined precisely. We know that there are two winter solstices per year, when the entire Halo is in constant darkness; each is followed by a warming period, as days start to occur and become longer and longer. If the maximum axial tilt is less than 75 degrees, the warming period ends in a summer solstice, the hottest time of year, and then the days start to shorten again and the Halo approaches the next winter solstice. If the maximum axial tilt is greater than 75 degrees, the hottest point is reached and passed and the Halo actually cools as it approaches its summer solstice. Then it heats up again, reaches the 75-degree point and cools back down toward the winter solstice. So the yearly cycle is either: Winter, Summer, Winter, Summer, or Winter, Summer, Mild Winter, Summer, Winter, Summer, Mild Winter, Summer.
Of course, this conjectural seasonal pattern is useless if precession makes the Halo's axis of rotation too variable. But I think there's hope that this is not in fact the case. The two primary causes of precession are the Halo's orbit around the Lagrange point, and the Lagrange point's orbit around Threshold. The second orbit is likely to be in roughly the same plane as Threshold's (to judge by our own solar system, where almost all orbits of natural bodies are nearly coplanar), and so the precession caused by it will tend to accelerate or decelerate the seasonal progression but not to disrupt the pattern itself (just as Earth's precession speeds up the seasonal cycle slightly, so that the equinoxes come a little earlier every year). The first orbit could very well disrupt the pattern, as its plane could be pretty much anything, but it's unlikely that the angular momentum associated with it is very large; to ensure long-term stability, both the orbit's radius and the speed of the Halo along it will probably be fairly small. Consequently, the precession from this source will hopefully be quite slow, and will not affect the seasonal pattern of any particular year.
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Wind Patterns
Though each season affects the entire Halo simultaneously, the sunward hemi-ring (that is, the ring you get if you cut the Halo in half along its rotational plane, analogous to the northern or southern hemispheres of Earth) in any particular season will be warmer than the spaceward hemi-ring. This should drive a system of sunward/spaceward winds, which will be the dominant feature of the Halo's weather system. At ground level, the winds will sweep sunward; near the sunward wall the air will rise and then sweep back spaceward at a high altitude, then sink at the spaceward wall. Due to the Coriolis force, the high-level spaceward winds will probably move counterspinward a bit; the ground-level sunward winds may or may not have a spinward component; its magnitude will probably be less than that of the counterspinward component of the high-altitude winds, due to friction with the ground.
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