Up to this point, things were fairly calculable. Now the Fi-part of SciFi starts - I draw a collection of landmasses underneath the climate lines. The distribution should be Earth-like (more water than land) since the Albedo was similar. From there then follows a measure of geology, and ultimately the different ecological niches, but fist, as promised, let's talk about weather.
My arguments here will be more qualitative as I seriously do not have the resources to run a full weather code for a fictional world.
The basic pattern is that air underneath the subsolar point heats and rises. This sucks air from the ice-covered dark hemisphere inward while the warm air reaches the upper atmosphere and migrates outward.
The massive inflow of air at ground level leads to a spinning vortex due to angular momentum conservation (that's a common pattern observed in objects as diverse as bathtubs and black holes), aka we get a super-cyclone. If the air at the subsolar point is moist, the water condenses, releasing latent heat and leading to copious cloud formation and rain - which provides substantial energy to fuel the process - the mechanism is really the same as for tropical storms on Earth.
Now, as seen above, the whole process is self-quenching, because the cyclone clouds have a high albedo and cool the subsolar point - which means that the highest energy is suddenly next to the subsolar point where there are no clouds and Mime can warm the sea undisturbed.
So the cyclone will generally go to where it can suck most energy from the sea, and hence 'wobble' around the subsolar point. In addition, because the orbit is eccentric, it will weaken at apoapsis simply because the available energy is less - so a winter pattern will look more like this
Likewise, the process of cold air flowing in at ground level won't equalize with the high atmosphere outward flow easily, because the upper atmosphere has 1/5 of the density, so it's not easy to transport the same mass that is coming in. There will instead be boundary instabilities at the division line between warm and cold air, with fingers of cold air reaching inward while pockets of cold air being pushed outward - leading to the developments of frontal weather - cold fronts with massive thunderstorms and clear weather in their wake, warm fronts with plenty of snow on the ice shields - here's a coldfront coming in.
Now, the consequence of a massive coldfront is that there's a broad stretch of planet which has a few days of exceptionally fine weather - and if the coldfront reaches up to the subsolar point, there's a massive heating of the ocean and hence a juicy path filled with energy to fuel the cyclone - so it may react and use this energy - and thus detetch from the subsolar point and migrate outward - creating a far-reaching warmfront over the ice (and sucking cold air inward in other places.
Finally, just like a hurricane can spawn secondary storms at its fringes, it would be fairly normal that the super-cyclone generates secondary cyclones which migrate outward (they're not much restricted by Coriolis forces, as we've already established these are weak). Again these would move warm and cold air when they reach the boundary between ice and thawed land.
So that, in a nutshell, is what I believe the typical weather patterns on this world would be and why.
The consequence is a high variability of weather - when a coldfront moves in, temperatures can easily drop by 20-30 K within a few hours. Winds can reach hurricane-force quickly when a cyclone makes landfall. Precipitation is copious nearly everywhere, especially at the windward side of mountains. All of this shapes the terrain which will have deeply-cut water erosion created canyons. Every lifeform needs to cope with the potential of high windspeeds and sudden freezing.
I'm currently in a discussion with a fellow biologist tossing around ideas what plausible adaptions to these conditions could be for plants and animals, and (with the understanding that this gets more into the Fi-realm) I will eventually present some of that as well.