Can we develop a working asymmetric crypto message protocol without an asymmetric encryption algorithm?
Use DLIES, which is essentially Diffie-Hellman with an ephemeral sender key. Assuming you know the receiver's public key, that will cost no extra round trips.

The sender does:

(eph_sender_private, eph_sender_public) = Generate_Key_Pair()
shared_key = SHA-512(Diffie-Hellman(receiver_public, eph_sender_private))
ciphertext = Encrypt(shared_key, message)
encrypted_message = (eph_sender_public, ciphertext)


The receiver does:

(eph_sender_public, ciphertext) = encrypted_message
shared_key = SHA-512(Diffie-Hellman(eph_sender_public, receiver_private))
message = Decrypt(shared_key, ciphertext)


Where Encrypt is authenticated encryption build from AES and SHA-512. For example you could follow the RFC Draft Authenticated Encryption with AES-CBC and HMAC-SHA which includes test vectors to minimize the risk of your messing it up.

If you feel fancy, replace the simply hashing in the shared key derivation by a proper KDF, such as HKDF-SHA-512.

To authenticate the sender, sign the message using a long term DSA key, which should be distinct from the long term DH key.