This answer may be sloppy as I just woke and am going through the questions I may have missed over the night, I apologize and will clarify anything that you find confusing.

No, the Consolidation did not create the cause of the War of the Hundred Years, it simply made the means for it.

There were several key actions that Hawkwing made, some under the advice of Moerad, some on his own prior to acquiring his most trusted adviser.

One action that pushed the possibility to the forefront was of course sending his heir to lead the expedition of Seanchan, leaving the line of succession unclear.

Another action was trying to cage the Aes Sedai in Tar Valon. As it is well documented, he hated them to the end, even going so far as to reject Healing that would have saved his life.

While the life for the average person was better during his reign, it did leave the nobles in the wind, not knowing what to expect from the next erratic order the High King was going to make.

With an unclear line of succession what was left of his relations at his death turned on each other, as well as anyone who wish to claim the throne of High King.

From the sources I have read for the cause of the start of the War of the Hundred years, it did seem mostly like unclear succession was the reason, as well as relying on his ta'veren ability to control the nobles instead of truly delving into the politics, was the main catalyst for the War to begin.