Concerning where the empire must be placed on the planet:
We know that the empire is at one of the "poles", but we don't know whether that means magnetic or axial. Since Rashek moved the planet too close to the sun, the best place to put the people is wherever the least energy from sunlight is incident on the surface, or geometrically, where the surface of the planet is facing as little towards the sun as possible. The magnetic pole can be more or less anywhere, so it would not really affect this property, and so I'll assume that we are at the axial pole.
The only data we have regarding the climate, aside from the ash which is artificial, are: the mists take over earlier in the third book in the outer dominances; and there are burnlands surrounding the empire, which are obviously due to the sun scorching the earth there. The burnlands must be in a region that is essentially facing the sun straight-on, whereas the outer dominances should receive less sunlight farther into the day than Luthadel, since sunlight banishes the mists. Honestly, these two facts are inconsistent, since the outer dominances are adjacent to the burnlands. However, since the mists are not exactly natural, there might be another factor (like proximity to the Well of Ascension) that repels them from Luthadel, so I will discard this datum.
We also know that the day lengths are normal. If they weren't, Brandon would have to have told us, because it would have a major effect on how people live: for example, the activities of Mistborn, who are out at night, or Vin's parties in the first book, which span a period of months but always take place at night. Not to mention that the mists being out "all day" wouldn't be so much of a phenomenon if during the winter, day were essentially absent. That means that whatever region the Final Empire is in, it doesn't really change its inclination towards the sun over the course of a year (much like where most of us live, which is at a middle latitude that tilts enough to give seasons, but not enough to give really wacky hours).
One solution is for the planet's axial tilt to be zero, with the Final Empire somewhere around Greenland: very high up, less of the sun reaches the surface, but the days are all exactly twelve hours long no matter the season -- of which there wouldn't be any, since there is no tilt. Oops. We could give the planet a very small tilt, restoring the seasons a bit; because of the more extreme heat of the sun, the seasons would be exaggerated to a more familiar level, without really affecting the day lengths. The burnlands are then in a wide band around the equator, which is delimited by some combination of less sunlight and more ash.
A highly eccentric orbit is another way to create seasons: the intensity of sunlight is proportional to the inverse square of the distance to the sun, since that's how the area of a sphere is. Also proportional to the square of the distance (roughly) is the area inside the planet's orbit. Kepler's second law says that the time taken to make a certain part of the orbit is proportional to the area swept out during that time, so to get x% more sunlight during the summer, winter has to be x% longer than summer. For someone at the approximate latitude of the northern US, where the seasons are pronounced, x is about 70, so for Scadrial to have seasons of normal temperature, their summers would be about 5 months long. It would also make the seasons uniform over the whole planet, while we know that Luthadel has milder seasons, and in particular, that the outer Dominances have snow, which seems to rule out the possibility that the ash is modulating the weather around the capital (since without the ash, on a hot planet the outer dominances would bake).
My conclusion: Scadrial has a small axial tilt. Not really that exciting, but it fits the facts.
