In method 1, you neglected the fact that, once a parcel of mass has left the container, it no longer receives heat. Only the gas that, at any instant of time still remains inside the container, receives heat.Okay, so assuming that this is what happens then, what is wrong with the 2 solutions I gave above?
I don't understand what you did in method 2. Certainly, if the temperature of the gas in the container is increasing, its internal energy E is increasing. After that step, I'm lost.