Step 1: Analyze statement (i).
Aerobic heterotrophic bacteria obtain both carbon and energy from organic matter (since they cannot fix CO$_2$).
Thus, statement (i) is correct.
Step 2: Analyze statement (ii).
Heterotrophs cannot use CO$_2$ as their carbon source. Only autotrophs fix CO$_2$.
Thus, statement (ii) is incorrect.
Step 3: Analyze statement (iii).
Aerobic autotrophic bacteria (like nitrifying bacteria) fix CO$_2$ as their carbon source and use reduced substances (like NH$_3$, NO$_2^-$) as their energy source.
Thus, statement (iii) is correct.
Step 4: Analyze statement (iv).
Autotrophs do not rely on organic matter for energy; instead, they use inorganic sources or light.
Thus, statement (iv) is incorrect.
Hence, the correct choice is: (i) and (iii) are correct.
Final Answer:
\[
\boxed{\text{(A) (i) is correct; (iii) is correct}}
\]