How do nodes verify Bitcoin Transactions?
Yes, but that doesn't mean that the network has accepted it.
There are rules for accepting a transaction.  Miners only include blocks that don't break the rules.  The rules include checking that the inputs are valid, that a coin isn't double-spent, that the output isn't more than the input, etc.


A miner can choose to include a transaction or not.  A miner could even choose to include a transaction that breaks the rules!  But if he does that, other miners that are following the rules won't accept his block and he won't get his mining reward.  So he has incentive to follow the rules.