No they are a star empire that spans a significant region of space and they are uber paranoid as a species and constantly on the look out for infiltration by their enemies the odds they wouldn't see as supernova from any point in their area of control and report it to their homeworld are nil.
The star was 150 light years away it would take the effects of that nova no less than 150 years to reach the homeworld and due to physics not all effects would arrive at once.
But in the movie it ignores physics which for a startrek movie is a major sin, it's science fiction but it was always based on science fact and went a long way to be believable.
Also on the red stuff we don't know it was in anything but glass because it's never explained let alone addressed.
I'm going to play devil's advocate because it's fun (compared to work).
"No they are a star empire that spans a significant region of space and they are uber paranoid as a species and constantly on the look out for infiltration by their enemies the odds they wouldn't see as supernova from any point in their area of control and report it to their homeworld are nil."
Maybe. But then maybe they wouldn't - when they're monitoring to detect intruders aren't they looking out for warp signatures? Perhaps the signs of stars going supernova aren't something their continuous scans are configured to detect.
"Also on the red stuff we don't know it was in anything but glass because it's never explained let alone addressed."
We don't know that their ship's floor is anything but plastic/metal, but I think we reasonably assume there's some sort of artificial gravity being generated. Even in the old, much more scientific days, they didn't need to explain every single piece of technology.