This is exactly what happens to me too, I dont believe it has smth to do with graphics, i tried already with minimum settings and normal, the gameplay is just not smoth.
Sometimes i click to move and its very slow to respond and start moving, althou moving is not the worse issue.
I right click on enemy unit or demigod to attack it and most times nothing happens, other times my demigod moves on top of the target and then starts attacking and only a few times he attacks as he sould.
With skills, like the fireball, i press shortcut to skill and click on target, sometimes he casts it nicely, other times he moves on top of the enemy target and only then casts it, other times it just wont cast it. i try multiple times but he just wont cast, i have to attack other target to get rid of the bug and cast fireball again.
On attack-move, i click attack-move in the ground at spot X, the demigod starts moving and starts attacking first creep in range, most of the times after killing the creep or creeps in range he lost the attack-move to spot X and just stays there, u have to order it again, and again, almost everytime he kills someone.
Another thing i thing could be greatly improved is the targeting system, what i mean is, i want to cast a targetted skill and some times, when there is lost of confusion, it get really hard to click on the correct target, even thou you have your mouse right on top of it, for instance Rook, that is not small, it happended to me several times in a row, just because he was standing next to a tower, the skill would always go for tower and not Rook.
I hope i could explain what i experience, will look more into this and hope to be more especific and explanatory next time.
Please keep this thread going because this is really gamebreaking, like someone before said, i dont mind because is against AI, but this is very bad for a competition game.