While carrying out a flame test on a compound W, a bluish flame is observed and on heating W, the compound decomposes producing a brown gas X and another gas Y, that rekindles a glowing splint.

A blue flame is characteristic of copper ions.
The cation in compound W responsible for the blue flame is $\boxed{\text{Copper (Cu}^{2+})}$.

b) (ii)

A brown gas is typically nitrogen dioxide.

A gas that rekindles a glowing splint is oxygen.
Gas X: $\boxed{\text{Nitrogen dioxide (NO}_2)}$
Gas Y: $\boxed{\text{Oxygen (O}_2)}$

b) (iii)

Compound W contains copper (from the blue flame) and decomposes to produce nitrogen dioxide and oxygen, which is characteristic of a metal nitrate.
Compound W is $\boxed{\text{Copper(II) nitrate (Cu(NO}_3)_2)}$.

b) (iv)

When heated, copper(II) nitrate decomposes into copper(II) oxide, nitrogen dioxide gas, and oxygen gas.

c) (i)

The formation of an immediate white precipitate upon adding $BaCl_2$ solution indicates the presence of sulfate ions, as barium sulfate ($BaSO_4$) is insoluble.
The likely ion present in solution is $\boxed{\text{Sulfate ion (SO}_4^{2-})}$.

c) (ii)

Effervescence producing a colourless gas that decolorises acidified $KMnO_4$ indicates the presence of sulfur dioxide ($SO_2$) gas, which is produced from sulfite ions ($SO_3^{2-}$).

Gas evolved: $\boxed{\text{Sulfur dioxide (SO}_2)}$
Ion present: $\boxed{\text{Sulfite ion (SO}_3^{2-})}$
Test: $\boxed{\text{Pass the gas through acidified potassium dichromate(VI) solution; the orange solution turns green.}}$

c) (iii)

The gas commonly used in fire extinguishers is carbon dioxide ($CO_2$).

Gas evolved: $\boxed{\text{Carbon dioxide (CO}_2)}$
Test: $\boxed{\text{Pass the gas through limewater (calcium hydroxide solution); it turns milky/cloudy.}}$