The unsaturated ether on acidic hydrolysis produces carbonyl...

Acidic hydrolysis of vinyl ethers involves the addition of water across the double bond followed by the elimination of alcohol/phenol to form carbonyl compounds. In the given reaction:
$CH_3-CH=CH-O-C(CH_3)=CH_2 \xrightarrow{H_3O^+} P + Q$
The hydrolysis occurs at both vinyl ether linkages.
The left part $CH_3-CH=CH-O-$ hydrolyzes to form propanal: $CH_3-CH_2-CHO$ (P).
The right part $-O-C(CH_3)=CH_2$ hydrolyzes to form acetone: $CH_3-CO-CH_3$ (Q).
P is an aldehyde (propanal) and Q is a ketone (acetone).
Fehling solution is used to distinguish between aldehydes and ketones. Aldehydes (except benzaldehyde) reduce Fehling solution to give a red precipitate of $Cu_2O$, while ketones do not react.
Lucas reagent is used for alcohols. 2,4-DNP reacts with both aldehydes and ketones. Saturated $NaHSO_3$ reacts with most aldehydes and methyl ketones, so it is not the best choice for distinguishing these specific compounds compared to Fehling solution.