H-NMR Equivalent Protons and Coupling of Glycerol
This might be confusing since both ends of the molecule are the same. The difference is that the alcohol points to only one side of the molecule.



So, while the hydrogen atoms on carbons 1 and 3 on paper have the same environment, they are not chemically equivalent when looking at the whole molecule.

Concerning your second question: The chemical shift of protons on heteroatoms such as Oxygen, Sulfur, or Nitrogen is highly dependent on the recording conditions (concentration, temperature, solvent) of the NMR. Frequently, the signals are broadened and show no coupling (broad singlett). The reason for this is the rapid intermolecular proton exchange. If there is a rapid exchange between two chemically non-equivalent groups within a molecule (like in this one), only a single, averaged signal is observed.