For your first question, assuming the air was dry to begin with, yes the water vapor will increase the total pressure.
Yes, if the cylinder were constant-pressure rather than constant volume, it would expand dramatically when heated. That's because in addition to the usual PV=NRT, you have an increasing N due to the evaporation of the water. At some T, the vapor pressure of water plus the pressure from the other gases will equal the set pressure. At that point, the container will expand until all the water has turned to vapor.
So long as the vapor pressure is less than the total pressure, it is only weakly affected by the total pressure (for fixed temperature) because the gas is nearly ideal and the chemical potential of the liquid is rather insensitive to pressure.
What you may perhaps be noticing is this: In a fixed-pressure container, ther'es a very sharply defined boiling point. It's the temperature for which the vapor pressure equals the total pressure. If there's no other gas around, as a function of T the system abruptly switches from all liquid to all gas. We've discussed that in some old answers. If there's some air around, we've described the behavior above. For a fixed volume container, there's a broad vapor-liquid coexistence regime even if there's no other gas around. Assuming the volume is bigger than needed to hold the liquid, the pressure just becomes the vapor pressure, until all the liquid is evaporated.
(published on 02/09/2011)