known_hosts has nothing to do with public key authentication. It's a list of servers you've previously connected to and fingerprints of their SSH keys to help verify the connection to the server is not being man-in-the-middled.
authorized_keys is a file stored on the server containing the SSH public keys of users authorized to log in to the server.
id_rsa is the private key.
id_rsa.pub is the public key. The public key is encoded according to the format in RFC 4253, but basically it's a PEM-encoded blob that contains the key type, length, and the values
Public key authentication does not involve passwords at all. There's no "password confirmation" step at all. When the client connect, it offers the fingerprints of keys it has to server. If the server can use one of those to connect, it asks the client to sign a request provided by the server (more or less some opaque blob of data). The client takes this request, signs it with the private key, and then sends it back to the server. The server verifies the signature of the blob, which shows that the client possess the correct private key (matching one of the public keys in
authorized_keys) and grants access.