The idea of the small dog retrieving sticks of dynamite is probably inspired by a classic internet urban legend, the nature and history of which is discussed here on Snopes.

Ray: New Romantics. Bunch of suburban poofs with doyleys on their heads.

Alex: Will you just shut up and listen to me! This is my bloody fantasy, and I will be listened to!
Gene: Excuse my colleague, education of a toff, manners of a sewer rat.

Gene: (to Alex) Take that seatbelt off! You're a police officer, not a bloody vicar.

Alex: A real, living, breathing Thatcherite businessman. How completely brilliant.
Gene: Personal friend of the Great Handbag herself, so try and behave.
Alex: I promise not to twang his red braces. More than once.

Gene: We found a small amount of explosives on land near the Royal Docks.
Alex: Well, technically a small dog found it. He's an even smaller dog now.

(After Alex reads off the anticapitalist screed from The Pop Group, "We Are All Prostitutes")
Gene: I'm sure Barry Manilow has covered that one.

The general response of David to Gene tossing the supposedly dynamite laden bag is incorrect for anyone actually experienced with dynamite. Though a high explosive, dynamite is exceedingly stable (that was, in fact, what made it such an important development and Alfred Nobel so rich). It will, in fact, burn much like wood (sawdust is commonly a significant ingredient as a matter of fact), and cannot be inadvertently set off by "a hard knock". In general it can only be detonated via blasting caps, which are themselves a high explosive only subject to explosion based on triggering by electric current. In other words, the shock of a minor blow, such as dropping it, will not set dynamite off. About the only possible justification here is the idea that the dynamite might be very, very old, and even that notion is somewhat specious.

George O'Dowd: Welcome to The Blitz.
Alex: Hey, thanks, George!

George O'Dowd, aka Boy George, an 80s persona central to the band The Culture Club ("Do You Really Want To Hurt Me?"), was a regular at, worked as a cloakroom attendant at, the trendy London nightclub The Blitz in the early 1980s.

Ray: New Romantics. Bunch of suburban poofs with doyleys on their heads.

The New Romantic movement was an early 80s musical movement from the UK to soften the harshness of the Punk Rock development, essentially the UK version of Emo, but much more enduring than 80s Emo, as it still has a notable following (though Emo has survived as a genre in itself, unlike New Romanticism). It included many bands such as Culture Club, Adam and the Ants, The Human League, Ultravox, Duran Duran, Visage and Spandau Ballet. It is generally classed as a sub-part of 80s New Wave music. Some of the members (notably The Human League and Spandau Ballet) were also called "Fashion Bands" due to their emphasis on neat tailoring and a clean, upscale look, often while still retaining elements of their punk origins.

Alex: (quoting from a 45 cover's lyrics) "We are all prostitutes. Everyone has their price. And you, too, will learn..."

From 1979 release titled (duh) We Are All Prostitutes, by the post-punk band The Pop Group.

(After Alex reads off the anticapitalist screed from The Pop Group, "We Are All Prostitutes")
Gene: I'm sure Barry Manilow has covered that one.

Barry Manilow was a 70s pop Icon most famous for very "non-punk" pop songs like "At The Copa" and "Mandy".

Gene: Would you like to take some advice from The Jean Genie?

The Jean Genie is the title of a 1972 single by David Bowie. It was the lead single from the album Aladdin Sane, and is one of Bowie’s most well-known songs. Gene has used the term as a self-reference previously in Life On Mars.

Gene: Careful, Bowles. That formica was hewn from the hills above Florence...

Presumably a reference to Alex as Sally Bowles, a central character in Christopher Isherwood's short novel called Goodbye To Berlin. The novel was later adapted into a Tony-winning stage play and then a film, called I Am A Camera (leading to the satirically termed review by critic Walter Kerr, "Me no Leica"). The title is quoting the novel: "I am a camera with its shutter open, quite passive, recording, not thinking." The novel, play and film were later the inspiration for the now classic stage musical and subsequent Oscar winning musical film directed by Bob Fosse, Cabaret .

Gene: Careful, Bowles. That formica was hewn from the hills above Florence...

In a sense, the Sally Bowles connection to I Am A Camera is appropriate though distorted, given that Alex perceives all of her experiences as some kind of drawn-out hallucination which she is watching and taking part in "observing, recording", but, in fact, thinking as she believes it to be a puzzle she has to figure out. This referent also provides a connective element back to Life On Mars, where Sam connects his existence to being like the Wizard of Oz, and Gene regularly refers to Sam as Dorothy, the central character of The Wizard of Oz. Ashes to Ashes is thus connected to I Am A Camera in the same sense. There are stylistic element to the visual look of the show, too, in Alex's costumes being somewhat over-the-top like that Sally Bowles wore in the film version of Cabaret.