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?
Nodes store a mempool of valid transactions. An invalid transaction can never be in a block, so invalid transactions never make sense to store or relay around the network. You can make invalid transactions with invalid signatures, but it won’t be accepted by anything.