What is the temperature of evaporating water (not boiling)?
No, the average thermal energy doesn't need to be the one corresponding to 100C.
Solo water molecules can exist in vacuum or in a gas phase with whatever kinetic energy. Liquid water condensation starts when the concentration of the water molecules is enough high (there is a tipping point), i.e. when the partial pressure of water reaches the vapor pressure for the given temperature.

From the classical thermodynamic point of view (similar argument to Hexacoordinate-C's comment), there is a chemical potential of water in the gas phase which is dependent on the partial pressure of water, and there is a chemical potential of condensed water for a given temperature. When the partial pressure is low, the gas phase chemical potential is lower than it would in the condensed phase -> it is stable in gas phase. When the partial pressure reaches the vapor pressure, the chemical potential is equal to the gas phase and condensed phase.