Rate my idea: is that a stronger GPG-like algorithm?
Your idea is no stronger than simply having a common shared password $P_1$ from which the symmetric encryption key is derived.


  If Alice encrypts a message with Bob's hashed password, even if someone knows the shared password, only Bob can decrypt the message


You assumed the hash of Bob's password – $H(B_1)$ – is public, so if Eve knows both it and the common password $P_1$ she can calculate $H(B_1)||H(P_1)$. She doesn't need to know $B_1$ itself.

A program can be written to require the password $B_1$, but Eve could write her own program that accepts either the hashes or the passwords.

I.e. unless you assume $H(A_1)$ and $H(B_1)$ are private, they do nothing. If you do assume that, they effectively become part of the shared secret.


  Only Alice knows the common password, so Bob's is sure that the cypher is surely made by Alice (no MITM attacks, fake identity like in GPG)


Yes, but the problem of sharing the common password $P_1$ remains. If you can securely exchange that password, you can likewise securely exchange GPG keys or any other encryption keys.