I have the exact same problem (in-game freezes when mouse is not moving). My mac is Mavericks (OS X 10.9.5).
I heard that people with Sierra can fix this problem by un-checking "shake mouse pointer to locate" option.
However, Mavericks does not have this "shake mouse pointer to locate" option to un-check.
I also find that the game works fine in Borderless mode, however, the graphic resolution becomes bad compared to the Fullscreen mode.
Riot has to fix this issue. I know that this in-game freeze has started since the 7.1 patch, and it has been almost 3 months. And, they still have not fixed this issue.
Not sure why they cannot come up with a solution.