Pressure is determined by the height of a column and the density of the fluid in a column: $p=\rho gh$.
Why the pressure at $P_1$ should be same as the pressure at $P_2$?
If we use dimensions in your diagram, they would not be the same.
What we can say about the first tube, is that the pressure at the bottom of the two columns will be the same. If that was not the case, there would be a horizontal gradient of pressure, which would cause fluids to move until the pressure equalizes.
If so, we can state, that the weight of the mercury column is equal to the weight of the water column. But the fraction of mercury column above the line is much smaller than the fraction of the water column above the line, so their weights and therefore the pressure at the level of the line would not be the same: $P_2$ would be greater than $P_1$.
Why the pressure at $P_3$ should be same as the pressure at $P_4$?
Because the columns of the water on both sides should have the same height (since the horizontal pressure at the bottom of the columns should be the same) and, therefore, equal fractions of the left and right columns will be above the line and, therefore, they will create equal pressure at the level of the line.
Why the pressure at $P_1$ should be same as the pressure at $P_3$?
It is hard to tell, since, in general, the drawing is not accurate (considering that the density of mercury is $13.6$ times greater than the density of water), but if you understand the answers to the first two questions, you should be able to figure that out for any given dimensions.