when you make a turn to the left, the right aeliron goes down thus making the extra angle of attack, thus creating the more lift, while the left aeliron goes up
is this correct?
Spot on!
pushing the nose down, causes the elevator to deflect upwards
is that correct?
Lets think about this one. Push the yoke (or stick) forward, nose pitches down. Why? Because you've done something to make the tail pitch up. To do that you would've made the elevator deflect DOWNWARDS which causes the tail to rise.
left rudder pedel causes the rudder to deflect to the left (assuming you are still facing the front of the aircraft)
Again, spot on (except pedal is the correct spelling).