A precise physical argument against the existence of an Ice-9-like state of water is probably not trivial (ab initio methods required?). However, there is a simple but powerful argument against it; we haven't found anything like it yet. Water is a very common substance, and humans have been applying the scientific method to it for hundreds of years, humans have manipulated it for hundreds of thousands of years, but most importantly, the entire Earth has been churning massive amounts of water around for billions of years. If an Ice-9-like state of water did exist, we should have seen it by now.
To be clear, superficially this argument may seem to be a fallacious argument from ignorance. However, we can qualitatively back it up with a bit more knowledge. Assuming that Ice-9 is the true ground state of water, and that all liquid water on Earth is currently in a false ground state, separated only by a kinetic barrier, and realising that the conversion of liquid water to Ice-9 is autocatalytic, we must ask ourselves, if it only had to happen once, why hasn't it happened yet?. The only reasonable conclusion is that, if it can happen, then the kinetic barrier to conversion must be enormous. The interconversion of diamond to graphite is a chemical process with a famously large kinetic barrier (around $\mathrm{540\ kJ\ mol^{-1}}$), yet we have plentiful evidence of it happening many times in geological time scales (and we can do it in the lab too!). Thus the kinetic barrier for conversion between liquid water and Ice-9 would have to be even larger, which is unlikely. There's also a limit to how high we can expect chemical kinetic barriers to be, which we can put very generously at $\mathrm{100\ eV \approx 10\ MJ\ mol^{-1}}$. Water molecules on Earth have had many, many, many interactions at energies way higher than this, as they have been incessantly exposed to cosmic radiation, nuclear decays, lightning strikes, high temperatures, etc. In short, the collective history of all water molecules on Earth has explored a vast region of the space of allowed states, and not once has an Ice-9-like state been found. It is thus fair to assume it simply does not exist.
Regarding your second question, I just remembered a great example: pure tin. At ambient pressure, solid tin can exist in two different polymorphs, and the transition between the most stable form is conveniently close to room temperature. Above 13.2 ºC, tin is most stable as a ductile metallic material, called white tin, whereas below 13.2 ºC, the most stable form is a brittle non-metal called grey tin. It is possible to sustain white tin below 13.2 ºC for a period of time, creating a metastable system, much like a supersaturated solution. However, an adequate disturbance will cause the white tin to fall to its ground state. Like Vonnegut's Ice-9, this transition is autocatalytic; once conversion begins, there's no stopping it below 13.2 ºC, creating the so-called tin pest. There are actually several great videos showing the process, which you can easily find on YouTube, such as this one.
