Marek's answer is rather too non-statistical classical. Your cork analogy looks to me at first sight not statistical in a way that gets close enough to quantum theory. As an aside, it's generally better to make your models slightly more abstract-idealized mathematical — note that Marek has replied with a model that is both much more abstract and much more idealized than your model.
If you're willing to make the potential well have expected depth $E$ with some finite variance because the apparatus that constrains the classical particle is noisy, however, then the analogy is not so bad, if the potential well is then sometimes shallow enough for the classical particle to escape. There is noise in quantum theory —insofar as one carefully thinks in terms of quantum fluctuations and is careful to distinguish them from thermal fluctuations—, so it doesn't seem unreasonable to introduce noise into classical models.
You can alternatively surround the classical particle with some kind of noisy environment, in which case the energy of the particle will fluctuate. If the fluctuations are enough that the energy of the classical particle is sometimes more than the depth of the potential well, then again the particle can escape from the potential well with some finite probability. If you haven't previously looked at Nelson models, get hold of his "Quantum fluctuations / by Edward Nelson, Princeton University Press, c1985."
I don't know of and couldn't find a web resource that gives an adequate description of this approach, however, even though it's arguably closer to being acceptable to Physicists than de Broglie-Bohm type approaches.
Personally, your intuition at this level is OK, but it's not in this context that the really difficult issues arise, and also not in the double-slit kind of context. Kochen-Specker and the violation of Bell inequalities are much more challenging. In response to those challenges I have found it much more productive to characterize the differences between classical random fields and quantum fields, both of which are intrinsically statistical, instead of trying to think in terms of classical particles in comparison with quantum particles. The random field context is close enough to the quantum field context that discrete energy levels are not a novelty. I presume you're not getting my Answers to your Questions, but your Questions are very familiar to me.