Arm patting as metaphor for patronizing conduct
Jean says to Phoebe: "It's just that I know exactly how Uncle Bill patted your arm-- just in the same way as he'd wait on the man at Christmas when he was in the Army. So democratic, so charming, and so English." Jean despises her Uncle Bill, who discreetly paid her expenses while she was in college and who (together with the retired Billy) has been discreetly helping Archie's entire household get by financially. She believes Billy pities Archie and his dependents, and she is correct. The patting of Phoebe's arm to reassure her after the street children vandalize his car during a visit is a metaphor for what Jean believes to be a superficial and distant approach to Archie's family.
In reality, Bill is doing the best he can to help Archie and his family without judging them. He is sick of bailing Archie and his family out financially and receiving only arrogant belligerance in exchange: he's good enough to be asked for money or used for financial aid, but he's not considered good enough to be invited to family celebrations such as the one planned for Mick's safe return. Furthermore, he has family of his own to support and must accordingly distance himself from the consequences Archie's financial irresponsibility. His ongoing aid is, in part, what has enabled Archie to continue wasting money and resources in his doomed music-hall. Everybody is quietly aware of this but nobody says so.
One of the reasons Jean rejects Graham Dodd is because she sees in him a future copy of Bill. She does not want to become like him, and although she moved away to London she does not want to turn into the kind of person who needs to distance herself from her family in order to protect herself from the consequences of their actions, and be bitterly resented by them for it. She'd rather be the one to feel the resentment.
Brother Bill as a "really big pot"
In one of her drunken rages, Phoebe describes Brother Bill, Archies elder sibling and the only other child of Billy Rice, as being "a really big pot". By this, she means a pot of something (she doesn't say what) with no flies on it. He is important, impressive, and apparently without significant flaws except that he "can't understand" Archie and his family. It's not that Bill can't understand them: it's that he disapproves of their bad decision making and does not wish to enable it.
Bill is a successful barrister who has raised three children, all of whom are well educated and have made good marriages. His youngest has married a military officer who is the son of a wealthy industrialist, and Bill and his family have successfully become part of England's modern upper industrial and professional class. Meanwhile, Archie (who received the same education and opportunity as Bill) is struggling in the music-hall business and cannot pay his bills.
Although Phoebe likes Bill because he is "a gentleman" who treats her kindly, Jean believes Phoebe likes him simply out of class admiration. On another level, both women bitterly resent the fact that Bill is successful financially but Archie is not, and she hates the fact he is distant. They fail to see how it is not Archie's poverty that drives Bill away, but the decision making of Archie and the people around them.
Singing as a metaphor for happiness
Old Billy Rice says, early in the play, that he hasn't done much singing of late. Formerly an entertainer and showman, Billy has an outstanding voice and is still a competent performer. But when he sees what the entertainment industry has become, with televisions in the pubs and naked women onstage at Archie's music-hall, he is disgusted. He therefore withdraws and does not go about to sing.
Hymn singing as simile for disapproval
When Billy refuses the drink Archie is trying to press on him, Archie says: "you look as though you're going to sing a hymn." Archie believes that his father is passing judgment on him and expressing disapproval. Billy is simply tired and wants to go to bed, and he is not interested in participating in the ongoing drunken carousing that Archie and Phoebe's life has become.
As dim as a bucket
Archie describes himself, facetiously, as being "as dim as a bucket" such that he doesn't realize when he's being insulted. He actually does. He is perfectly aware of the harm he is doing to the people around him (since he is an intelligent, well educated person who is perfectly capable of picking up on slight social cues). He is aware of the pain he causes them and of their resulting disapproval and withdrawal, but throughout the play he covers up for it by acting obnoxious. He uses the simile preemptively to stave off the legitimate criticism of others.
Fertilizer business as metaphor for dishonesty
During his last monologue, Archie says that "we're all in the fertilizer business now". What he means is that everybody, at some level, is misrepresenting himself or herself to advance an agenda. The "fertilizer" in question refers to bull manure.
