I am also interested if there are any differences in behaviour when the ratio A/B changes.
A liquid-vapor interface is not a static interface, there is a so-called liquid-vapor equilibrium where molecules in the liquid phase are continuously escaping from the liquid into the vapor phase and vice versa vapor molecules are continuously captured by the liquid. In equilibrium, the number of molecules leaving the liquid into the vapor and leaving the vapor into the liquid are equal.
The liquid-vapor equilibrium is a function of pressure and temperature. When you heat a liquid more molecules will leave the liquid into the vapor (due to higher kinetic energies) than molecules from the vapor phase into the liquid and the interface is no longer in equilibrium; the liquid phase will slowly vaporize until it has disappeared. When you increase the pressure, the molecules require more kinetic energy to leave the liquid phase because the liquid molecules are closer together and their interaction energies are higher. This effectively increases the boiling point of the liquid and it takes a higher temperature to evaporate all the liquid (e.g. in pressure cookers).
The opposite effect occurs when you decrease the pressure, i.e. you lower the boiling point of the liquid. It requires less energy (lower temperatures) to vaporize all the liquid. So by using a vaccuum pump you will first suck out some of the vapor reducing the pressure, causing the liquid to start vaporizing, creating vapor which is again sucked out by the pump. Only after all the liquid has vaporized, will the production of vapor stop and will you be able to reach a vaccuum.
So to answer your questions; If the situation can be created, some of the liquid will vaporize to fill section $A$ and the liquid will reduce in mass. The ratio $A/B$ dictates if only some of the liquid is vaporized or all of it.
The liquid will vaporize. You can just keep sucking. Will not be able to create a vacuum until all the liquid is gone.
If you had a barrier between the two and created a vacuum and then removed the barrier you would have vapor of the vapor pressure of the liquid at that temperature.