It is not really clear to me what is the role of the miners after they mined.
If I understood correctly a miner has also the task to "verify" any new transaction and for doing so he automatically gets a fee....is that correct?
Can anyone explain that to me?
Indeed, the miners (should) verify new transactions before including them into a block and also before accepting a new block as valid. However, they do not get fees for verifying transactions, they only obtain the fees if they mine the block where the transactions are included.
From the bitcoin wiki:
The transaction fee is processed by and received by the bitcoin miner. When a new bitcoin block is generated with a successful hash, the information for all of the transactions is included with the block and all transaction fees are collected by that user creating the block, who is free to assign those fees to himself.
Therefore, the role of the miners after they mined is just to keep mining (and so to keep validating new blocks when they receive them, and new transactions before including them into blocks).
