Answers
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A full node holds the whole version of the blockchain, it’s connected on the network, waiting for new transactions to add on the mempool or new blocks to add on the chain. Simplified Payment Verification just queries a full node to have some information about the status of some transactions. Usually we can use SPV on the mobile phone.
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A wallet signs new transaction with the private key and controls that we have some UTXOs to spent. A full node receives this new transaction, saves its into the mempool and broadcasts to other full node and miners on the network. So, when a transaction is broadcasted, it means that is shared between nodes in the mempool.
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Miner chooses from the mempool the transaction with highest fees and composes a new block. Every miner works on a different block of transaction. They compete each other to solve the algorithm puzzle behind the hash code, trying to guess a nonce number. When this puzzle is solved, the new block is added on the chain and broadcast at the node on the network. To be sure there will not be stale blocks on the chain, for accepting new payment transactions the network wait the next six blocks mined.