ChaCha20: is this a potentially safe way of using nonce derived from private key if the key/nonce pair is never reused, for a given plaintext?
I realize in ChaCha20 the nonce should be random and unique each time…


Nope!

For the ChaCha stream cipher, it is not safe to choose the nonce at random.  At 64 bits long, the ChaCha nonce is too short to be chosen at random.  In general, you should use a sequential message number under any single key as the nonce.

For the XChaCha variant, like the XSalsa20 variant of Salsa20, it is safe to choose the nonce at random because the extended nonce is 192 bits long so the collision probability becomes nonnegligible only after an unimaginable $2^{96}$ messages.  But your question was about ChaCha without an extended nonce.


  Could the nonce be entirely deterministic and derived from the key so long as the key is only used exclusively once for a distinct plaintext (akin to the one-time-pad). I realize this is risky if the key/nonce pair is re-used on a different plaintext, but wouldn't it be as safe as the one-time-pad if the pair is never reused on a different plaintext message?


If the key is chosen independently and uniformly at random for each message, then you can safely use the fixed nonce zero.  Don't mess with hare-brained schemes like copying bits of the key around into other places.


  plaintext: 68656c6c6f
  
  resulting chacha20 ciphertext: 702da4b13e682d4b840b19cde305583b415765f936


Something is wrong with your use of ChaCha because your ciphertext length is not exactly the same as your plaintext length.  [Update: This is because the question is actually about the ChaCha/Poly1305 authenticated cipher of RFC 7539, not about the ChaCha stream cipher originally described by Dan Bernstein with a different nonce size.]



Also: Write down what your protocol is and how it fits into your application so that a reviewer can confirm you're following the rules of ChaCha.

Also: Don't just use ChaCha and leave it at that.  Use an authenticated cipher (or AEAD, authenticated encryption with associated data) like crypto_secretbox_xsalsa20poly1305.