Are bond angles constant?
No, bond angles are not constant. For example, the low-energy IR absorption of water (at $1595 \:\mathrm{cm^{-1}}$) is from the H-O-H bend

There are a few techniques to measure bond angles. For polar compounds, microwave rotational spectroscopy can often be used on small molecules in the gas phase.

I'd guess though, that most bond angles are measured from crystal structures using X-ray diffraction.

Keep in mind that spectroscopic techniques always have a timescale associated with them. So any experimental measurement is time-averaged. In the case of bond angles, while the angle will change, the average value is the equilibrium - at some points, the vibration will create a larger bond angle, and at some points in time, the bond angle will be smaller.