Step 1: Read the clues backward.
$S$ is a hydrocarbon made from $Q$ by soda lime. Soda lime removes one carbon as carbonate, so $Q$ must be a carboxylic acid and $S$ has one carbon less.
Step 2: Use the Grignard clue.
$P$ plus ethyl magnesium bromide, then acid, gives an acid $Q$. A Grignard reagent adds to carbon dioxide and after hydrolysis gives a carboxylic acid with one extra carbon. So $P$ is $CO_2$.
Step 3: Build Q.
Ethyl group ($C_2H_5$) joining $CO_2$ gives $C_2H_5COOH$, that is propanoic acid. So $Q = C_2H_5COOH$.
Step 4: Apply the HVZ reaction.
$Q$ with $Cl_2$ and red phosphorus undergoes alpha halogenation (Hell-Volhard-Zelinsky), putting $Cl$ on the carbon next to $COOH$. This gives $R = CH_3CHClCOOH$.
Step 5: Apply decarboxylation.
$Q$ with soda lime loses $CO_2$, leaving the hydrocarbon with one carbon less than the acid. $C_2H_5COOH$ has three carbons, so $S$ has two carbons, giving ethane, $C_2H_6$.
Step 6: Collect the answer.
So $P, Q, R, S$ are $CO_2$, $C_2H_5COOH$, $CH_3CHClCOOH$, $C_2H_6$.
\[ \boxed{CO_2,\ C_2H_5COOH,\ CH_3CHClCOOH,\ C_2H_6} \]