Step 1: Checking statement (A).
Carbon shows negative oxidation states (–4 in CH$_4$) as well as positive oxidation states like +2 (CO) and +4 (CO$_2$). Hence, this statement is correct.
Step 2: Checking statement (B).
Among group 14 dioxides, CO$_2$ is the most acidic due to its small size and high electronegativity of carbon. Therefore, this statement is correct.
Step 3: Checking statement (C).
$^{13}$C is a stable isotope of carbon. The radioactive isotope of carbon is $^{14}$C. Hence, this statement is incorrect.
Step 4: Checking statement (D).
Carbon does not have vacant d-orbitals and cannot expand its octet, so its maximum covalency is four. This statement is correct.
Step 5: Conclusion.
The incorrect statement is option (C).