I've read some articles about the bitcoin network, some of them say it is necessary for a transaction, before adding to blockchain, to be validated by more than half of all nodes in bitcoin network. Is my understanding true? If yes, is it true to say that after exactly 51% of nodes have validated the transaction, it will be added to blockchain?
It is not the percentage of nodes that matters, but the percentage of total network hashing power. If 51% of the network's hashing power is mining on top of a block, then you could say the network considers all the transactions in that block to be valid, and accepted. Other nodes will continue to validate the transactions independently, but that is so they know whether or not to trust the blocks that the miners propagate. It allows them to be trustless.
So what happens when 51% of hashing power is mining on top of a block containing a transaction that 49% of the network consider to be invalid? This is what a hard fork is. It means the 49% will continue mining on what they consider to be the last valid block.
No, it is not necessary for any node validate a transaction before it is included in a block and thus added to the blockchain. A transaction can be included in a block without being validated, but doing so risks the transaction being invalid and thus making the block invalid. This has happened several times in the past; miners will make their own transaction and include it in their own block without broadcasting it to the rest of the network.
In bitcoin blockchain, a miner would first validate transactions like making sure that the senders have enough balances in their accounts before adding the transactions to a block. Then the miner again will find the nonce of the block that will produce the desirable hash. After that, the miner will broadcast the block into the network. Other miners in the network will validate the transactions and hash, and if OK, they will mine on the top of broadcast block by that implicitly signaling (voting) that is valid. Otherwise, the block will be simply ignored by others miners. There is no need in bitcoin for 51% of the nodes to validate block. PoW assumes that 51% of the hash power is with honest miners, and then they will end up producing the longest chain which means actually that there is some kind of an implicit voting among miners. By mining on the top of a valid chain, you are implicitly voting for it.