Let's say user A has public key P and corresponding private key P'. Can someone else initiate a transaction for public key P with some wrong private key, let's say P''? Once this goes to the memory pool, miners will see that it is not signed with correct private key P' and will reject the transaction. Does it happen this way? Or is it that I can't initiate a transaction at all with wrong key?
You can initiate a transaction with the wrong key, but it will actually get stopped before the transaction gets to the miners.
All Bitcoin transactions are reviewed by the Nodes, to make sure they are sending valid amounts of Bitcoin and were signed by the correct private key. If there is anything wrong with the transaction (wrong signature, not enough BTC), the network will reject it and probably ban you for 24 hours.
This is a fairly confusing topic because most crypto content focuses on explaining the miners, and very little focus is spent explaining the nodes.