Why both leds does not light equally when power is applied?
I'll address only the title question of why the LEDs do not light equally given the following circuit. I have duplicated your actual circuit below. 







simulate this circuit &ndash; Schematic created using CircuitLab

When V2 is high, C1 charges through D3 and the current through D3 due to the capacitor rapidly drops to near zero. D1 is then essentially in parallel with D2+D3. Since D3 provides some voltage drop D2 will be much dimmer than D1. the only way it would be of similar brightness would be if D3 was ideal (0V drop) and it is not (though it's a Schottky so it drops very little- if you replace D3 with a regular diode such as a 1N4004 you will find the brightness difference even greater). 

When V2 goes to zero, the voltage at R1 drops and D3 and D1 are reverse-biased, so current flows only through D2 and R1 in series. 

You should be able to run the simulation with my shown circuit and see how it behaves as you change parts. If you split the resistor R1 into two, as you've found, it will be much closer in brightness, though one will still see a bit more current than the other. You can even make one resistor a bit lower than the other to equalize the currents at a given input voltage etc.