Would this authentication on a Diffie-Hellmann-key-exchanged channel work?
In the question as it stands now, nothing prevents Eve to relay Do you have a cat? to Bob, get the answer A = Yes by testing A for Yes or No against hash(e_bob+A) obtained from Bob, then send hash(e_alice+A) to Alice and defeat the protocol.

In a now-gone comment, it was added that we


  should use "strong values" for A; only Q is sent, plain A is never sent.


Notice that then, in order for Alice to perform a check, A must be known to Alice in advance. If this "strong value" (or even the convention to use it the way described) is known to Alice and Bob, but not Eve, then the technique works.

If only the "strong value" is unknown to Eve, it acts as a secret key in an ad-hoc protocol (for something academic we'd want to use this "strong value" as key to HMAC of message e)

If the convention is unknown to Eve, that's not cryptography in the modern sense of Kerckhoffs's second principle; and that's brittle, and can't be generalized.