AND Gate Not Working Properly in Combinational Circuit
I applaud your willingness to experiment however you need to learn a little more about how transistors work,


Give each LED its own resistor to limit the current through them remove R1.
Use your NPN transistors on the ground side of the circuit to make their behaviour easier to predict.
Use resistors in series with each transistor base something around 1k should be suitable.


Hopefully from this you can rethink your circuit design. You may want to check the truth table for out1 as well if this is intended to be a one bit adder.