In Bitcoin, any transaction that confines to the consensus protocol is valid. This is unlike other ledger based system like Ripple, where the validators vote on whether the transaction is valid and if it should be included in the ledger. In those ledger systems, the validators gets reward for helping maintain the ledger.
When your Bitcoin full node receives a transaction, your node will check through all the protocol requirements. This includes verifying that the inputs consumed in the transaction have not been spent in the past (i.e. no double spend), there is a valid signature that unlocks the bitcoins, inputs are greater than outputs as so forth. You can find a detailed list of the transaction verification rules here.
Does the Engine verify everything what is coming thru and i just have to pay for Power, Online Time, Memory and so?
That is correct. As mentioned above, when Bitcoin Core will receives a broadcasted transaction, it automatically verifies it against the consensus rules to determine that they are valid. Incorrect transactions are discarded and not relayed to other nodes. You only need to have your machine running with the software and it would take care of it.
Why my Budget didn't move a Single Satoshi up or down
In Bitcoin, only the miners get the block reward and the transaction fees associated with it. A full node's job is to ensure that the block which the miners have mined conforms to the protocol requirement. If say a miner turns malicious and mines a 'fraud' block, the full nodes connected to the network will reject that block and will not relay it further. Other miners also assume that the nodes on the network are honest and do not build further on the 'fraud' block mined. Thus the malicious miner not only loses the block reward and transaction fees, but also has spent his electricity and mining power to waste. Thus full nodes play a very significant role in the Bitcoin network to ensure everybody is acting in an honest manner.