Let's suppose that a miner mines a block and adds it to the blockchain. When it gets the reward ? I suspect that first of all the other nodes have to verify that its proof of work is correct and it's not cheating. Right ?
Miners don't receive rewards from anyone else - they give the reward to themselves as the output of the coinbase transaction, which includes the block subsidy and any fees from the txs in that block.
They then broadcast this block to everyone else. Each node that receives the block independently validates it to ensure it meets consensus rules (has a valid proof of work, does not contain any invalid transactions, has the correct subsidy and fees charged, etc.).
If any node believes the block is incorrect, it will reject it and continue to mine or wait for another valid block at the same height.
If the block is considered valid, it is added by each node to its own copy of the blockchain. Once 100 blocks have been mined extending the block in question, the coins in the coinbase output are considered mature and may be spent by the miner of the block.