Checking this set miners validate that UTXOs included into given transaction exist and are unspent.[/quotes]
Nodes do this, not miners.
Technically, only the 80-byte block header is hashed via SHA256 with the nonce, not the entire block.
I wouldn't say the block header is hashed "with" the nonce; rather, the nonce is already part of the block header. And of course it is actually hashed via SHA256 twice.