When does a transaction end up in the transaction pool?
The transaction pool (or mempool, as it's usually called) is not a network-wide pool. Each node maintains its own mempool.

When a node receives your tx, it will validate and add it to its own mempool, and possibly broadcast it further on. In some cases, such as a lot of unconfirmed txs, a node's mempool may be full, in which case it will not be added.

There is no network-wide consensus required to add your transaction to the mempool, each node does so individually, regardless of how many other nodes have validated that transaction.