Deeper questions about Ethereum Consensus and Execution Layers

I'm trying to better understand the current connection between the Consensus Layer (PoS Chain) and the Execution Layer (PoW Chain), so that I can move on to better understanding the Merge.

I understand that the Consensus Layer is running parallel to the Execution Layer right now. What I don't understand is how that works exactly, so hereby my questions:

• How are the Consensus Layer and the Execution Layer running parallel? Whats the connection? Is the Consensus Layer simply copying the Execution Layer? Are there even blocks on the POS chain already?Is the total supply of the consensus chain token equal to the execution chain supply?
• Where are the current staking rewards on the consensus chain coming from? Is it additional inflation?Are these rewards visible on the execution layer already?