Regarding the two objects, I think is a case of "dativo superfluo", which some people think are not really indirect objects, but an "objeto of interés"
Indica el interés de la acción, es decir, no es el destinatario sin más (Complemento Indirecto), sino el afectado por la acción.
Las expresiones con se me / se te / se nos / se os implican que el suceso queda fuera del control de la persona y que ésta se ve afectada por ello
Meaning that the object is there not really to tell you who suffers the action, but to stress who is affected by the action instead. It seems that there are several different kinds of dativos regarding not only who suffers the action but the action suffered (please, check the link for more exhaustive information).
Se te va el avión the event of the plane leaving affects the person addressed by this sentence. Compare
Se va el avión
which has a meaning like
The plane is leaving with
Se te va el avión
which means "Your plane is leaving", you better hurry up, because there are consequences for you. You are going to be directly affected by this event
That is why we have two objects there that seem redundant, one is pointing who suffers the consequences of the action.
About the differences between
ir, think about this:
The plane is leaving, leaving you behind.... The first
leaving would be from
Ir since the plane goes from here to somewhere.
Dejar, when talking about traveling, has the connotation of "dejar atrás". You could say
El avión deja Madrid (para dirigirse a Barcelona)
El avión deja la pista y sube hacia al cielo.
I understand your confusion, because the plane is actually leaving behind the person you are addressing, but then you would be saying
El avión te deja en tierra (the plane leaves you behind)
Or to phrase it with the words of the example:
A las 3:30 se cierran las puertas y si no estás dentro, se te deja en tierra (you are left behind)
So, as you can see, with
dejar you will need a complement for that verb. Also is obvious that you suffered the consequences of the action, so there is no need of the "se te" construction to indicate it. In
El avión te deja en tierra you are de CD of the action done by the plane. If you use
The plane leaves you may (or may not) use the construction to indicate the consequences.