There are two things at work here - gravity, and gas pressure. In the beginning, due to gravity, denser air is at bottom and lighter air is at the top. You may ask why it is this way to begin with and the answer is in the word "denser". At every level, there is an equilibrium density in the beginning and it is higher at the bottom and lower at the top. When some air near bottom is heated, it does not push up, all it does, is expands in all directions due to its increased temperature (and so increased pressure). Due to expansion, its density goes lower. And due to this lowering of density, the density equilibrium is disturbed. Then gravity brings back the density equilibrium by pulling dense air down more than it pulls the hot air. So hot air only expands, gravity does the rest.

Now you may ask why gravity pulls denser air more than the lighter air. Because denser air has more mass per volume and so more gravitational force per unit of volume. (GMm/(r*r)).

Therefore in actuality, even though it is dense air that is pushing down (due to gravity), the hot, less dense air has no where to go except up, and so it appears it is pushing up but it really does not (or we can say it pushes in all directions, not only up, due to its increased pressure). The cold air moves down only from sides, it can not move down from directly above because of increased pressure of hot air.

Air does indeed flow from high pressure to low pressure area (see the wind arrows on a weather chart), but in the case of two rooms the much more important effect is that of warm thinner air rising towards the ceiling when the air from the two rooms gets mixed.

Thus, cold air from the cold room will be leaving the room close to the floor (if the temperature difference is large enough you can actually feel it, otherwise you can use a candle to detect the direction of air movement). At the same time, warm air from the warm room will rise and move into the cold room close to the ceiling (again, sometimes you can feel this otherwise you can detect it with a candle). Similar air movements take place between your house and the outside when you open your house's door in winter or summer.

The tendency of warm air to rise towards the ceiling is actually exploited in floor heating. This was understood and taken advantage of already by the ancient Romans, see hypocaust.