When lead azide detonates, you get elemental lead and diatomic nitrogen. So yes, the Pb++ oxidizes the azide.
When acetylene detonates, you get a mixture of organic compounds. But I don't believe that there's any redox involved. You just get various hydrocarbons with single and double carbon-carbon bonds. And just to be clear, this is when there's no oxidizing agent present.
In chemistry, an oxidising agent (oxidant, oxidizer) is a substance that has the ability to oxidize other substances — in other words to accept their electrons. Common oxidizing agents are oxygen, hydrogen peroxide and the halogens.
The jargon being that an oxidiser is an substance that can accept electrons, and doesn't have to by oxygen.
Just take oxidizer to me an “something that accepts electrons, like oxygen would”. Of course, there’s some oxidizers that can accept electrons from things that oxygen can’t (think of fluorine, for example) but... it’s just a term.
There’s also an enormous continuum of timescales, all the way from the detonation of high explosives, to deflagration of low explosives, combustion of fuels, to... sedate rusting, which is just oxidation of iron or steel. YMMV.
When lead azide detonates, you get elemental lead and diatomic nitrogen. So yes, the Pb++ oxidizes the azide.
When acetylene detonates, you get a mixture of organic compounds. But I don't believe that there's any redox involved. You just get various hydrocarbons with single and double carbon-carbon bonds. And just to be clear, this is when there's no oxidizing agent present.