transactions – How does the mining network know/agree when a block has been defined and thus needs to be mined? (The hash determined.)

The network does not need to agree on what the next block looks like, until it has been mined. Miners act independently to create new blocks, each miner can (and probably will) include slightly different transactions, in a slightly different order, in the block templates that they are mining on. In fact, a miner could fill a block up entirely with otherwise unbroadcast transactions, that no other node on the network would have knowledge of before that miner published a valid block containing said transactions to the network.

The blockchain record is the method by which the network comes to independently agree on the specific ordering of transactions. Until the time at which a new, valid block is added to the chain tip, nodes on the network will not know what the next block will look like.