Bitcoin transactions validation process
Everything MCCCS wrote is good, but just to clarify on your last question:


  How many nodes must approve/verify the winning hash before it is accepted into the bitcoin network as the next block to be added?


This question is slightly misguided: there is no 'accepting a block into the bitcoin network', because this assumes there is some network-wide threshold of acceptance that must be reached, or some sort of central yes/no switch for a block's acceptance.

The reality is that the network is made up of independently acting nodes. Each node decides for itself whether or not a block is valid, and it is by these 'consensus rules' that the network happens to come to agree on what the longest (most work) blockchain is. Once a valid block is broadcast, each node will independently work to verify that it is valid, and then add it to their local copy of the blockchain if it passes all the tests. The network is designed such that we expect all nodes that follow these consensus rules will remain in consensus, even though they are acting independently.