thermodynamics - What actually makes boiling happen?

Boiling can be breezed over easily with a few rudimentary diagrams and a couple equations, but I seek a deeper explanation.

The definition of boiling is that the vapor pressure in the liquid is equal to the vapor pressure of the air. This seems reasonable in open containers because when pressure exceeds 1 atm, then bubbles can form. However, consider a closed container at lets say 40 degrees celsius. This is too cold for boiling, but the vapor pressure of the liquid must be equal to the pressure exerted by the vapor (system will move until this equilibrium is reached). Why doesn't this water boil? The pressures are the same so bubbles can form and boiling can occur.

My fundamental misunderstanding of boiling at the molecular level leads to more related questions:

When heat is added, why does the temperature rise until boiling point and then all the energy goes towards breaking bonds? In melting water, the vapor pressure of the water and solid are equal so they can both coexist. If this is the case, isn't this technically the triple point? because water, solid, and vapor are present? (This can't be of course because melting point is not equal to triple point).