Step 1: Analyse Statement I.
Primary alcohols are oxidised by Jones reagent \((\mathrm{CrO_3/H_2SO_4})\) to the corresponding carboxylic acids.
\[
\mathrm{CH_3CH_2CH_2OH}
\;\xrightarrow{\mathrm{CrO_3/H_2SO_4}}\;
\mathrm{CH_3CH_2COOH}
\]
Hence, Statement I is correct.
Step 2: Analyse Statement II.
Alkyl bromides are not directly converted into carboxylic acids using only CO and \(\mathrm{H_3O^+}\). Carbonylation requires specific catalysts and conditions.
Hence, Statement II is incorrect.
Step 3: Analyse Statement III.
Aldehydes are readily oxidised by alkaline potassium permanganate to carboxylic acids.
\[
\mathrm{CH_3CHO}
\xrightarrow{\mathrm{KMnO_4/OH^-,\ H_3O^+}}
\mathrm{CH_3COOH}
\]
Hence, Statement III is correct.
Step 4: Final conclusion.
Therefore, the correct statements are
\[
\boxed{\text{I and III only}.}
\]
Hence, the correct option is \(\boxed{(C)}\).