A lot of it obviously depends on the manner in which the war is concluded.
Let me shoot a few holes in your questions:
How will the populace react to martial law and the removal of democracy in favour of direct military rule?
The GTVA does not have the military strength to do a full occupation of Sol.
Will the council of Elders be killed off / disappeared to a remote research facility for a future of brain dissection?
Knowing that, why would they kill off the one set of authority figures that is actually respected across Sol space? Wouldn't doing so increase the risk of a civilian uprising (one that would be impossible to control, and would be very likely to spread to other GTVA systems)?
Will the Fedyaeen hand over Cassandra to the GTVA?
Again, this depends on the manner in which the war is ended. Personally, I would find it unlikely.
Will the change over from trade to war production cripple the economy of the Sol system or boost it - bearing in mind that the Terrans are dependent on the Vasudans to support their own systems.
Sol has an enormous raw economic power, but there is little that can be converted into military production quickly; Retooling and rebuilding the shipyards, as well as retraining the workforce, will take a few years, years in which the economic boost will be felt in other areas first.
Will we see forced exoduses from Sol as the GTVA tries to stabilise the colony world populations?
See above re: Inciting rebellions. The GTVA, while certainly more authoritarian than other governments, is not an absolute dictatorship.
Besides, if your goal is to contain Ubuntu as much as possible, spreading citizens who have experienced no bad side effects from living under Ubuntu all over your area of influence strikes me as a bad idea.
All in all, the GTVA you describe is villainous to an extreme we (as the BP team) did not intend. The GTVA is harsh, yes. It will make some decisions that the Sol populace may not agree with, yes. But at the same time, there is no benefit to courting disaster by behaving like a bunch of jackasses towards a population that is 1. the same size as the GTVA's, 2. Concentrated in a single system, and 3. able to coordinate.