If (hash mod N > K), then the miner will move to the next block without calculating any hash.
It is impossible in any decentralized system. You could have some difficulty and accept only blocks with hash modulo 2 equal to zero, that would be as hard as mining with doubled difficulty, you cannot expect that people will stop mining just because your official mining software is doing that. If N is some natural power of two, then it is the same as taking some last bits of the hash. By checking blocks for the classical Proof of Work and some condition with hash modulo N, you are just rising the difficulty in some obscured way.
Yes. But like in Bitcoin, we bet that the network will be controlled by honest miners who will play by the rules. And even if they don't, their work will be simply ignored by the honest nodes and hence waste of money for the miners involved in such practice.
the policy stipulates that cars whose license plate ends in an odd number are only allowed to circulate on odd-numbered dates and vice versa
It works, because license plates are distributed in centralized way.
Yes. There is no way to stop a miner from hashing if he want to launch his machine. But technically his hash work will go to dustbin.
The replacement of $nonce$ in PoW by $sign(nonce)$ will force the miner to either share its private key with the mining pool or calculate the hash itself.
No, it won't change anything. Miners will share their private keys with mining pools. Today they are mining directly to the pool's address, so sharing the key requires the same level of trust as today. But that's just one option. In any useful coin you can still make transactions. That means you can mine a block where you send your reward to yourself, but you can include a transaction sending some older coins to the pool. Then, the pool can accept your shares only if you paid them first, in this way you can reach the same system as you have today, but just with more transactions to obfuscate what is going on (if for example you will find a way to punish miners for sharing keys).
Indeed. But it reduces pool mining to be only between miners who can share their private keys with each others. A total trust is required. Which means that there a is large number of miners who won't be able to create pools unless they trust each other 100%. And this in itself will create a limitation in the pool formation. It won't stop it.