What motivates the gang in "The Destructors", considering the house's age, leadership dynamics, the burnt banknotes, and their regard for Old Misery?

First of all, in terms of the age and beauty of the old house, the boys have no appreciation for it. They've grown up in poverty in the midst of a charred, post-war landscape pockmarked by the debris of war and destruction. To them, beauty is a totally alien concept, one that they've never experienced at any time in their short lives. So they don't look upon Old Misery's home as a thing of beauty; they just see it as presenting a great opportunity for some wanton destruction.

The boys of the Wormsley Common gang don't ask for much in a leader. They just want someone who can come up with innovative and exciting new ways to have fun. Enter Trevor. It's his big idea to reduce Old Misery's house to a pile of steaming rubble. The other boys don't hesitate to go along with his crazed plan. In doing so, they make Trevor, or T, as he's now become, their leader. The chances are that he will retain that position for as long as he can lead his merry little band to ever more outrageous heights of willful destruction.

Burning the banknotes is an expression of T's contempt for the life his family has been forced to leave behind. Now that he's come down in the world socially, he actively hates society and all it represents. As such, he has no respect for others' wealth or property. The other boys in the gang come from much poorer backgrounds, but it's a sign of T's charisma that he's able to get them to act against their class interests by burning money that could well have been very useful to them and their families.

The boys' consideration for Old Misery shows that they're much more interested in destroying things than people (at least for now). What makes T a particularly ruthless leader is the fact that he's able to act out of the purest, coldest logic. It isn't personal with T; he wants to get back at society as a whole—as symbolized by Old Misery's house—rather than any one individual.

What motivates the gang in "The Destructors", considering the house's age, leadership dynamics, the burnt banknotes, and their regard for Old Misery?

The gang's motivation for pulling down Old Misery's house seems to have nothing to do with any personal feelings any of them have for Old Misery himself.  Their only personal encounter with Old Misery prior to the destruction reveals their collective disaffection with adult authority; though Old Misery kindly offers them some chocolates, they respond with suspicion and ingratitude. In pulling down his house, the boys demonstrate no desire to physically hurt him or punish him. To most of them, it is just a house, and its destruction is no better or worse than the destruction of any other English house during the Blitz. 

Trevor is the exception when it comes to their motivation for destroying this particular house.  As the son of an architect who has recently fallen in the socially stratified world of England, Trevor recognizes the significance of a house designed by Wren. It is Trevor's decision to target this particular house, and in doing so, he leads the others in an act of socioeconomic protest.  The world that has rejected his family must suffer.  The boys simply follow Trevor's lead to satisfy their own appetites for wanton destruction. 

The burning of the banknotes underscores the idea that the boys, particularly Trevor, are not motivated by greed. They are not thieves. 

Their concern over the gang's leadership fades as the boys immerse themselves in the work of destroying the house.  Each boy concentrates on his own role and thinks less of who is calling the shots once.  Their motivations become personal and they develop agency in how they each exert their own destructive power.

What motivates the gang in "The Destructors", considering the house's age, leadership dynamics, the burnt banknotes, and their regard for Old Misery?

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.

What motivates the gang in "The Destructors", considering the house's age, leadership dynamics, the burnt banknotes, and their regard for Old Misery?

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.

What motivates the gang to destroy the house in "The Destructors"?

The plan to destroy Old Misery's house originates with Trevor ("T"), who supplants Blackie as the group's leader with his audacious proposal.  T's motivation stems from his anger at how his family has fallen in status after his father has lost his position as an architect. T recognizes that Old Misery's house was designed by noted architect Sir Christopher Wren, and it is a way for him to symbolically strike back at a world that has rejected his father. T's motivation is also his desire for acceptance into the gang, and wresting leadership away from Blackie is an additional benefit.

The other boys' motivations for destroying the house are much more mindless.  The destruction that came with The Blitz has desensitized them; they see the ruins of other houses littering their landscape. It doesn't occur to them that there is any real additional harm in their actions as they begin to tear the house apart. Although they don't have any fondness for Old Misery, neither do they have any particular hostility towards him.

What motivates the gang to destroy the house in "The Destructors"?

In "The Destructors," the gang's motivation for destroying Old Misery's house is three-fold.  First, the gang wants to raze the house to improve their prestige among the other street gangs.  Blackie and the others feel that such a stunt would bring the Wormsley Common gang recognition and respect. 

Second, the gang wants to destroy Old Misery's house because they distinctly do not trust Mr. Thomas.  Even after his attempts to ingratiate himself by giving the boys some Smarties candy, the gang members still eye him with suspicion, commenting that he probably found the candy or stole it. 

The final motivating factor for destroying the grand old house is to rebel against society.  The boys collectively detest what the opulent old home formerly stood for, which of course is the superiority of the upper class.  By tearing down the old house, the boys effectively declare their own independence and strength against the hierarchy of society. 

What motivates the boys to destroy the house in "The Destructors" by Graham Greene?

Destroying Mr. Thomas' grand old house is all Trevor's idea.  After all, he is the first one to 'scope out' the place as a potential target.  T presents the idea to the gang as an amazing opportunity to do something that no gang has done before:

‘They’d never know. We’d do it from inside. I’ve found a way in. We’d be like worms, don’t you see, in an apple. When we came out again there’d be nothing there – nothing but just walls, and then we’d make the walls fall down – somehow.’

The gang is motivated by their desire to prove themselves worthy to the task of pulling the house down.  The challenge of the physical task appeals to their creative, destructive talents immensely. 

The Wormsley common gang also wishes for fame and increased notoriety.  Blackie predicts that "the fame of the Wormsley Common car park gang would surely reach around London."  It is important to note that the boys are extremely resistant to doing anything that they perceive might land them in "the jug" like "pinching things;" the boys naively do not see their act of destroying the house as a criminal activity.