1. Identify P from cumene process

Iso-propylbenzene is cumene. On oxidation with $\mathrm{O_2}$ followed by acidic hydrolysis, cumene hydroperoxide gives:

$\text{cumene} \xrightarrow[\mathrm{H_3O^+}]{\mathrm{O_2}} \text{phenol} + \text{acetone}$

So,

$\mathbf{P} = \mathrm{CH_3COCH_3} \quad (\text{acetone})$

2. Identify \mathbf{Q} from reaction of acetone with 3 equivalents of chlorine

Acetone is a methyl ketone. Exhaustive $\alpha$-chlorination of both methyl groups with 3 equivalents of $\mathrm{Cl_2}$ gives:

$\mathrm{CH_3COCH_3} \xrightarrow[3\,eq]{Cl_2} \mathrm{CH_3COCCl_3} \;\text{or equivalent trichloro ketone?}$

But from the next step with $\mathrm{Ca(OH)_2}$, this must be the haloform-type cleavage precursor, namely chloral-related chemistry obtained through chlorination of acetone to give trichloroacetone:

$\mathbf{Q} = \mathrm{CH_3COCCl_3}$

Then with $\mathrm{Ca(OH)_2}$:

$2\,\mathrm{CH_3COCCl_3} + \mathrm{Ca(OH)_2} \rightarrow 2\,\mathrm{CHCl_3} + \mathrm{Ca(CH_3COO)_2}$

Thus,

$\mathbf{R} = \mathrm{CHCl_3} \quad (\text{chloroform})$ $\mathbf{S} = \mathrm{Ca(CH_3COO)_2} \quad (\text{calcium acetate})$

3. Check option A

$\mathbf{P}$ is acetone and $\mathbf{R}$ is chloroform. In presence of $\mathrm{KOH}$, they undergo the reaction used in chloretone synthesis:

$\mathrm{CH_3COCH_3 + CHCl_3 \xrightarrow{KOH} (CH_3)_2C(OH)CCl_3}$

This is chloretone (chlorobutanol), obtained after acidification/workup.

Hence statement A is correct.

4. Check option B

$\mathbf{R}$ is chloroform. Chloroform on oxidation by air/oxygen in presence of light gives phosgene:

$2\mathrm{CHCl_3} + \mathrm{O_2} \xrightarrow{h\nu} 2\mathrm{COCl_2} + 2\mathrm{HCl}$

So statement B is correct.

5. Check option C

If $\mathbf{Q} = \mathrm{CH_3COCCl_3}$, then with aqueous $\mathrm{NaOH}$ it undergoes haloform cleavage to give:

$\mathrm{CH_3COCCl_3 + NaOH \rightarrow CHCl_3 + CH_3COONa}$

It does not give $\mathrm{Cl_3CCH_2OH}$ and $\mathrm{Cl_3CCOONa}$.

Hence C is incorrect.

6. Check option D

$\mathbf{S}$ is calcium acetate. On dry distillation/heating, calcium salts of carboxylic acids give ketones:

$\mathrm{Ca(CH_3COO)_2 \xrightarrow{\Delta} CH_3COCH_3}$

Thus heating $\mathbf{S}$ gives acetone, i.e. $\mathbf{P}$.

So D is correct.

7. Final answer

Correct statements are:

$\boxed{A,\; B,\; D}$

This matches the stored correct answer.