Yes, yes and yes.
If you cool a spin glass rapidly (or, more likely to happen in a computer model: start it in a completely random state, which corresponds to infinite temperature, and immediately set the temperature to a low value) then it will indeed end up frozen into a higher-energy state than the ground state. This is known as "quenching." There would be no need for simulated annealing if this didn't happen - the purpose of the annealing is to try and avoid it.
In fact you usually end up in a "meta-stable" state - it will eventually decay to a lower-energy state, but this is likely to take a long time, because the system has to climb up a large energy hill before it can fall down the other side. In spin glasses the time taken to reach equilibrium can become infinite.
Making a small perturbation to such a state can indeed cause a rapid transition to a lower energy state. When there is frustration, flipping a spin can reduce the energy of one coupling while raising the energy of another. This can lead to a "cascade" or "avalanche" of many spins flipping one after the other. The dynamics of a spin glass at a low but finite temperature consist of multiple such avalanches, the size of which (I believe) obey a power law distribution.
Finally, something similar can indeed happen in the Ising model. Suppose you start the Ising model with a smallish positive external magnetic field and a high temperature. If you now reduce the temperature to a very low value, you will end up in a state where almost all the spins are aligned in the same way, with only a small probability of fluctuating in the other direction.
Suppose you now change the external field so that it has a smallish negative value. Since the model is frozen, nothing much will happen at first. Small domains of the opposite spin will tend to shrink to zero size, because the cells on the boundary of the domain are on average surrounded by more cells in the original state than the new one. However, you can note that the system is no longer in its ground state - it would have lower energy if all the spins were flipped the other way. It turns out that although small perturbations shrink, a large enough one will grow. So if due to a thermal fluctuation or an external perturbation you end up with a large enough domain of spins flipped the other way, it will grow fairly rapidly until it covers the whole system.
(All of this is assuming there is some sensible dynamics defined on the Ising model and the spin glass models, such as Glauber dynamics.)
I think this Ising model version is probably the best analogue to Prince Rupert's drop, since avalanches come in a variety of sizes whereas this phenomenon always affects the whole system.