RSA cryptosystem: Encrypting a signed message
Encrypts the resulting message/signature pair with the public key of Bob (the recipient): 
  Is that $P_B((M'', \sigma))$? What does this mean: $(P_B(M''), P_B(\sigma))$ or $(P_B(M''), \sigma)$?


Neither. It's simply $P_B(M'')$. The resulting message/signature pair refers to $M''$.