Why do guitar strings behave so nicely? To explain the harmonics on a guitar string, we use 2D models of the string. For example we assume that the string can only go up and down. But the string is inherently a 3D object and it could vibrate in a combination of side to side and up and down motions. My question is:
Why does this 3D problem reduce to a 2D problem?
My first idea was that no matter which weird 3D starting condition we set for the string (for example by plucking it in 2 different directions), it quickly settles to a state that can be described by a combination of the two groups of normal modes: up and down , side to side. Is this right?
Bonus question: In practice, if I pluck the string side to side, will it ever start vibrating up and down after some time?
 A: Given a very simplistic model of a string in space, fixed at both ends, you're right that the vibration can be in any direction, and furthermore that the vibration can be decomposed as a combination of vibrations in some basis (such as up/down and left/right).
However, due to the rotational symmetry of the setup, when the string is plucked in one direction (without any torque), it will oscillate in a plane, and hence it will resemble a 2D problem.
A: As to the extra question:

[...] pluck the string side to side, will it ever start vibrating up and down after some time?

I do rather expect that the plane of swing will veer.

For comparison: there are subtle oscillation effects in the case of trying to set up a Foucault pendulum.
The string of the pendulum setup is supposed to be equally flexible in all directions. But the upper attachment point must be very, very accurately rotationally symmetrical.
It is very hard to avoid a small difference, which manifests itself as follows: there will be a particular plane of swing with an ever so slightly longest period of swing and perpendicular to that the period of swing is shortest by ever so slightly a margin. (This type of difference is referred to as an anisotropy.) The pendulum bob is released randomly. So in effect the initial swing can be described as a linear combination of two perpendicular swings, with those two component swings having an ever so slightly different period. Because of that difference in period the plane of swing will open up; from a plane of swing it will develop into an ellipse shaped motion. With the plane of swing opened up you get momentum transfer from one component plane to another, the effect of that is a veering of the axis of the ellipse.
Naturally, the above is what tends to happen over the course of many cycles of the oscillation. A guitar string doesn't have a lot of sustain; you will soon pluck it again; maybe there isn't enough time for transfer to occur.
On the other hand, in the way that the guitar strings run over the saddle no provision at all is made to avoid anisotropy. So a difference in period depending on the direction of the plane of oscillation is to be expected, hence I rather expect that within the available time there will be transfer of momentum from one plane of swing to a perpendicular plane of swing.

(Anisotropy: an-isotropy. Isotropy is when things are the same (iso) in all directions.)
