Is is possible that a miner modified a unconfirmed transaction (like changing the output of transaction to the miner himself) and put it into the local block, and after 10 mins the miner luckily solved the PoW, putting the block on the chain and broadcast it to others?
No, because the transaction would be invalid. The vast majority of transactions contain digital signatures which signs the entire transaction. If the transaction is changed, the signatures will become invalid and thus the transaction will be invalid. And if a transaction is invalid, the block containing it is also invalid.
If a transaction was modified and finally linked onto the chain, how others check whether the tx was valid or not? (others only check if PoW answer was right or wrong, not checking tx, am I right?)
No, this is completely incorrect. ALL full nodes verify all transactions in all blocks that they receive (as well as transactions received outside of blocks). Just because a block has a valid proof of work does not mean that the block is valid. It must still build upon a valid block and must only contain valid transactions. Full nodes still verify that transactions contained within a block are valid.
Contrary to popular belief, miners do not say what transactions are valid. Their job is to determine the order of transactions, within certain constraints. It is the job of full nodes to verify transactions, and all miners (or the mining pools) should be running full nodes.