"So now they're looking at the possibility the nerve agent was administered through flowers and it has me wondering if these poisons ever die? Or do they just continue to exist? So if for example there was a trace of the poison on a grave stone would it always be there? Or would exposure to wind, rain, sun, snow etc eventually obliterate all trace of the poison"
Yes, it would eventually degrade. That aspect of nerve agent performance is its persistence - ie how long it exists in form that can do harm before it breaks down. How long it takes depends on the type of agent (as well as amount present) so a stable, persistent agent could stick around for much longer, months (you can add thickeners to ones which would otherwise evaporate rapidly), and you need specialist cleaning to remove it for sure (decontamination)
My guess is that the people who fell ill were in contact with the primary source, or a very strong secondary source. Also, they have spread it - quite possibly by touch - to at least one other location. Which means that every location they had visited since they left their house needs to be checked and cleaned.
It's not quite like being infectious - they wouldn't have been shedding as one would a virus. But say they still had some on their hands, and they touched things, that might be enough for transfer. Especially if it was readily water soluble and they were sweaty/damp for any reason