3) You are mistaken ... 
Look at what scheme a regular pool works:
Here, ordinary users are forced to look for a Hash in the WHOLE range of nonce values. And users who are connected to the pool search only in the selected segment. Hence the gain in the speed of finding a suitable hash and a greater total reward.
And now see how the pool will work if the POW algorithm is based on my algorithm:
Are you see? Ordinary users search EXACTLY in the same range of values as users who are connected to the pool.
So - there will be no gain from pool.
Look at what scheme a regular pool works:
Here, ordinary users are forced to look for a Hash in the WHOLE range of nonce values. And users who are connected to the pool search only in the selected segment. Hence the gain in the speed of finding a suitable hash and a greater total reward.
And now see how the pool will work if the POW algorithm is based on my algorithm:
Are you see? Ordinary users search EXACTLY in the same range of values as users who are connected to the pool.
So - there will be no gain from pool.
Based on your image, pool miners look for different value range, which means parallelization is still possible and there's gain from joining mining pool.
If it's wrong, then the image illustration is wrong.
From what i understand the chain is constructed in sort that there can be only one long sequence of operation for a given block height, so its not based on scanning a rangé of value with a statistical chance each round, but a deterministic set of rounds to solve the problem, and no partition can be computed in parrallel as each round dépend on the precedent one, it cant be made faster by breaking the work on different miners who can work in //.
I think the picture is wrong actually
its not even based on hash.