The difference is being engaged when you declare the walk action. There's a difference between being engaged and being in a mdoel's engagement range. For example, you can be in a model's engagement range, but if there is no LoS between the two models, then they aren't engaged. For a disengaging strike to occur three conditions must be met, the first is that the two models must be engaged. The second is that the model moving (model A) is inside of the engagement range of the model that isn't moving (model B). The third is that model A is attempting to move outside of the engagement range of Model B.
So as for the two scenarios you have, in the first scenario with the corner, Blue would have to use 1 action to walk outside of LoS, while still being in Red's engagement range, and then use their second action to walk out of engagement range. For the second scenario, as soon as Green is summoned in, Blue is allowed to move freely because LoS was broken, thus breaking their engagement.
As for your Hazzardous terrain questions, you have A B and C correct.