Question about button circuit
Sure. Take a multimeter and measure the resistance between the left-most pin and each of the other pins with all the different combinations of button pairs pressed. Write down the results.
Then build a voltage divider for each output (Connect a resistor to ground on each output, and then connect that pin to an analog input on the Arduino. The resistor value to use will depend on the readings you get, but a 1K ohm resistor would be a good starting point.)
Then set up the 2 input pins on your Arduino as analog inputs and measure the voltage with no buttons, top button, bottom button, and both buttons pressed for each pair. Write those values down. Then write code that checks for voltages roughly equal to your measurements for each button press combo (You will need check for ranges of values, since the values will vary some.)