energy when boiling

1. It is simply that the kinetic energy of molecules depends on temperature irrespective the molecules are in liquid (water) or in gas (steam) state.
Just imagine what would happen if the rms speed (or KE) of water molecules at 100'C were not equal to that of steam molecules at 100'C. No water molecules could escape from the boiling water and become steam molecules as the rms speed of steam molecules still not yet reached. Thus, 100'C would not be the boiling point of water. You need to raise the temperature of water to that beyond 100'C for its molecules to have higher rms speed in order to become steam. This, of course, is a contradicting situation.
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The KE of water molecules increases with increase in temperature, but I am in doubt if they are proportional.

Take a mass (m) of water for example, the rise of temperature (T) would indicate an amount of heat (Q) given to the water equals to m.c.T, where c is the specific heat capacity of water.
But the amount of heat Q is used to increase both the KE and PE of water molecules, i.e.
Q = m.c.T = increase in KE + increase in PE
Clearly, the increase in temperature T is not proportional to the increase in KE, as there is also an increase in PE.
In gases, your relationship holds, because there is no increase in PE on gas molecules when the gas temperature rises.
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2. Your equation is just the First Law of Thermodynamics, dQ = dU + dW
where dQ is the heat supplied, dU is the increase in internal energy of the substance (i.e. the increase in PE of molecules as there is no change of temperature during boiling, and hence no increase in KE), dW is the work done by the substance (in this case, it is the work done on expansion).

dU and dW are different. The former is work done against intermolecular force between molecules. The latter is an external work done against an external force (atmospheric pressure).