Are Context-Free Grammars with only one Production Rule always Unambiguous?
Unfortunately, your conjecture is wrong.

For instance $S \rightarrow aSSb | \epsilon$ is ambiguous.

To see that take $w: aabb$. For this string we have following two distinct derivation tree possible. In following derivation trees $e$ represents $\epsilon$

        S                            S
  /  /    | \                  / /    |    \
a  S      S  b                a S     S     b
 / | \ \   \                    |   / / | \    
a  S  S b   e                   e  a S  S  b
   |  |                              |  |  
   e  e                              e  e




Second counter example:

Consider grammar $S \rightarrow SaSbSaS | \epsilon$
It's easy to see that this grammar is ambiguous. (hint: take $w: abaaba$.)



Third counter example:

Consider grammar $S \rightarrow abSabSab | \epsilon$.
Take $w = abababababab$. It's easy to see that there are two distinct derivation trees.

Actually this grammar is directly derived from grammar $S \rightarrow aSaSa|\epsilon $ by homomorphism $h(a) = h(ab)$.