Why is the pressure on the poles higher in Summer and lower in Winter?
My answer will only relate to the question in the title: Why is pressure on the poles higher in summer and lower in winter?

Due to the axis tilt of the earth the winter hemisphere gets less heated by solar radiation. This leads to a high temperature gradient from equator to the winter hemisphere pole (strong baroclinicity). In contrast we have a low temperature gradient from equator to the summer hemisphere pole. 
The missing solar heating in the winter hemisphere additionally leads to descending air at the pole leading to low pressure at jet stream level (around 300 hPa). On a non rotating planet the air would of course directly flow from the high pressure (equator) to the low pressure (pole) to remove the pressure gradient. As warm air would flow to the cold pole air, this would also weaken the temperature gradient. But due to the rotation of the earth, air flows around the area of low pressure. This flow gets increasingly stronger as the meridional temperature gradient strengthens. 

This maybe seen from thermal wind balance, which tells us that the geostrophic wind $V_g$ is related to the temperature gradient $\nabla T$ by
$\frac{\partial V_g}{\partial \ln p} = -\frac{R}{f}k \times \nabla T$.
Here $p$ is pressure, $R$ is the gas constant for dry air, $f$ is the Coriolis parameter and $k$ is vertical unit vector. Assuming there is no zonal temperature gradient, we can conclude that a strong meridional temperature gradient leads to an increasing change of $V_g$ with height. We can observe this strong wind as the strong/weak jet stream in the winter/summer hemisphere. What does all of this have to with the pressure?

The jet stream acts as a barrier, dividing the warm and cold air masses. The Temperature will keep getting lower which will also increase the pressure and temperature gradient. In return the barrier (jet stream) gets stronger. This positive feedback mechanism remains until after winter solstice. Then the radiation reaching the pole will increase again, warming the air masses. Which will then lead to a weakening of the gradients.

So summarizing the above one could say, the missing radiation cools the air leading to a low pressure system at jet stream level. The jet stream itself gets stronger as the temperature gradient increases and acts as a barrier for warm and cold air masses and therefore maintains the pressure difference.