In answering this question, I will take into account each of your categories separately:
1: Age/Beauty of the house
The boys are living in a war zone in which bombing raids have and still are terrifying London. We are told that Trevor's home and some of the other boys' homes as well have been leveled by the bombs, making their lives chaotic and uncertain. Because Old Misery's house still stands despite its age, the boys, especially Trevor, become jealous at the injustice of its survival. Trevor, whose father was an architect, mentions “ 'It’s a beautiful house,' and still watching the ground, meeting no one’s eyes, he licked his lips first one way, then the other." He reacts to its beauty which spurs his need to destroy it.
2: Gang's Concern over Loss of Leadership
Young people are fickle. While Blackie had been the longstanding leader of the gang, Trevor, an older boy, seems more appealing to the rest. He had a more interesting agenda and an intricate plan. While Blackie was temporarily hurt by his pals' lack of loyalty, the idea of being left out overcame him, and he rejoined the group. His desire to be a part of something was more important than being in charge of it.
3: Burning of Banknotes
The burning of Old Misery's life savings is a key indicator that the demolition was about placing a type of rationality and logic in a highly irrational and illogical environment. The plan is to systematically destroy the house from the inside out "...like worms...in an apple...." Trevor makes it clear to Blackie that stealing the money would mean they hated Old Misery, and this job had nothing to do with emotion, just destruction.
4: Consideration for Old Misery
In keeping with Trevor's idea of keeping emotion out of it, the boys must also treat him as nicely as possible when he returns home early. They do lure him to the loo, but once he is secured inside, they treat him kindly. They bring him blankets and food. Again, the destruction is as unemotional as bombs are; they do not overtly wish Old Misery harm.
The devastation cause d by the war has left everything in ruins except for Old Misery. The last standing object of beauty is both miraculous and upsetting to the gang of boys. The loss of humanity fills the boys with a need to destroy it, to complete the ruination of their community.
Nothing has value anymore. Not the long-standing social order-respect for elders, or the leadership among the boys. Money has no value either, and burning it is symbolic in that aspect.
In some respects, the gang feels it is better to do away with all vestiges of the life before the war, and by destructing the house, they have accomplished that.