Time allowed to a node to react to verify a transaction and number of verifications?

Do the person acting as full nodes have to be active and online around for a particular decided time, like a job where you have to be around your machine because a transaction for verification can pop up anytime and if you are not around what happens then?

There's no need. If a full node uses the same rules to decide whether a transaction is valid as you do, it will always agree with you on transaction validity, so there's no need to wait for it to make a decision. If a full node uses different rules to decide whether a transaction is valid than you do, then you don't care whether it thinks a transaction is valid or not because it isn't part of the same network you are.

Also, the Bitcoin has more than 30000 nodes, how many verifications it actually need to become a part of ledger because how can you guarantee that this number of people will verify it in this fixed amount of time otherwise the wait could go indefinite?

If you think it's a valid transaction in a valid block that is part of the chain that you've accepted, then you think it's part of the ledger. People who agree with you on what makes a transaction valid will always agree with you on that decision, so you don't need to check with them. People who don't agree with you on those rules don't matter -- it's only the set of people who agree on the rules for transaction validity and block inclusion who can participate on a network together.

The secret to the way public blockchains work is that all transaction data is public, all state data is public, there are no administrative functions, and all participants agree on the rules for what makes blocks valid, what makes transactions valid, and what each transaction does. So you don't need to ask anyone else if something is valid and don't care about nodes that disagree with you on validity questions.

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.