Step 1: Analyse Statement I.
The central carbon atom in \(\mathrm{CO_3^{2-}}\) is \(sp^2\)-hybridised.
This statement is correct.
Step 2: Analyse Statement II.
The total negative charge of \(-2\) is delocalized equally over three oxygen atoms.
\[
\text{Average formal charge on each O}
=\frac{-2}{3}
=-0.67
\]
Thus, the magnitude of the average formal charge on each oxygen atom is \(0.67\).
Hence, Statement II is correct.
Step 3: Analyse Statement III.
The resonance hybrid does not contain one single bond and two double bonds. Instead, all three C--O bonds are identical with bond order
\[
\frac{4}{3}.
\]
Hence, Statement III is incorrect.
Step 4: Analyse Statement IV.
Because of resonance, all three C--O bonds have identical bond order and therefore equal bond lengths.
Hence, Statement IV is correct.
Step 5: Final conclusion.
According to the given options, the correct answer is
\[
\boxed{\text{II \& IV only}.}
\]
Hence, the correct option is \(\boxed{(A)}\).