Time allowed to a node to react to verify a transaction and number of verifications?
You seem to misunderstand how transaction validation and finality works.
Nodes validate transactions as they receive them in the mempool or in new blocks - these validations check for standardness (for mempool txs), and consensus compliance  (ensuring there are no double spends, valid utxos, etc).
However, a transaction is not final simply because a number of nodes have confirmed it. A transaction is final ONLY if a miner includes it in a block AND that block is accepted as valid by the network. This finality happens regardless of whether there is 1 node on the network, or 30000.
Note that even after being mined into a block, conventional wisdom says to wait for new blocks to be mined on top of that block, to provide a reasonable guarantee against minor forks rolling back a transaction you consider final. Most exchanges and service providers wait for 2-6 confirmations before considering a transaction truly final.