There is a node of sorts. But it depends on the mode of vibration. A given violin has a complete (and orthogonal) set of normal modes. This is true of any such bounded surface. Some will involve portions of the back going along with the top.
You can say nothing simple about any of this. It is not simple. It is analogous to quantum mechanics because in both cases one gets eigenfunctions with their eignevalues (frequencies of vibration.) If you cannot understand the nature of these normal modes, you should not talk so "qualitatively" about ANYTHING.
As for the rest of you, violin physics is not a mystery, it is just very complicated. Tap tones are the few lower eigenstates of the vibrating instrument. To solve the problem, one needs finite-element-analysis because of the strange and complicated shapes involved in a violin. Numerical analysis, in other words, is the only way to find anything about what is going on.
And one more thing, each normal mode will have characteristic damping. There are many kinds and models for damping, and this would make a numerical analysis much more involved.