Atmospheric Pressure inside a closed room

From Pascal's law, we know that pressure is isotropic, which means that at a given location in a fluid, it acts equally in all directions. So, at a given location, the horizontal force per unit area acting on a small vertical surface is the same as the vertical force per unit area acting on a small horizontal surface.

Usually, a room is not hermetically sealed, so it is not totally separated from the atmosphere. Any connection between the room and the atmosphere allows the pressure to equalize (by air flowing in or out). As we said above, pressure acts horizontally also, so air can come through a vertical crack just as easily as through a horizontal crack. In a house, there are typically vents in the attic which allow communication with the atmosphere.

If the room were totally hermetically sealed from the atmosphere, then you could impose any air pressure you wanted inside the room. It would not have to match the outside atmospheric pressure. But, the forces on the walls could get pretty large between inside and outside as a result of the pressure difference, and you would have to be pretty careful so that the room didn't implode or explode. When tornadoes occur, the atmospheric pressure outside drops substantially, and people are recommended to open the windows (to allow the pressures to equalize) in order to avoid the windows blowing out (or even worse).

This is a duplicate as far as atmospheric pressure goes.

As long as the container is not air tight there will come equalization of pressure. To understand why pressure equalizes one has to see the derivations of the ideal gas law, PV=RT using statistical mechanics, for example here. The attribute "law" is indicative of a thermodynamic law, which was observed to hold, not derived. Only after the understanding using statistical mechanics it could be derived.

Gas in an air tight container will keep the pressure it had when in equilibrium with the atmosphere unless temperatures change. The motions of the gas molecules exert an effective kinetic pressure on any surface they impinge on ( remember pressure is force over area) according to the ideal gas law.