(a) Sodium is kept immersed in kerosene oil because sodium is a very reactive metal that reacts vigorously with the oxygen in the air and with water. If sodium reacts with oxygen or water, it catches fire in an exothermic reaction.
(b) White phosphorus is kept immersed in water because it starts burning spontaneously when it comes into contact with air. But it does not react with water.
(c) No, we cannot keep sodium immersed in water because it reacts vigorously even with cold water to form sodium hydroxide, hydrogen gas and a huge amount of heat.
2Na(s) + 2H2O(l) 2NaOH(aq) + H2(g) + HEAT