Do chemical bonds have mass?

As far as the theory goes, you are absolutely correct, the (negative) binding energy between atoms in a molecule contributes to the total mass of that molecule, so a stable molecule is less massive than the sum of the masses of its constituent atoms.

However (as you yourself calculated), the mass difference is absolutely tiny, and as far as I know, it has never been measured. But the principle is no different from the mass deficit that occurs in nuclear reactions and that, in turn, is readily measurable. Consider the atomic mass of deuterium ($2.01410178\,\rm u$) vs. helium ($4.002602\,\rm u$), which is about $0.64\%$ less than the mass of two deuterium atoms. The difference is the energy that would be released in a fusion reaction.

So yes, in general, removing energy from a system decreases its mass, and conversely, adding energy to the system increases its mass. The most extreme example perhaps would be protons and neutrons: roughly $99\%$ of their masses come from the (positive) binding energy between their constituent quarks, and only about $1\%$ is attributed to the quark rest masses.

Yes, bonds have mass, like every other kind of energy.

This can be significant; if you had a glueball (a hypothetical particle made of massless gluons), it would have mass, and all of the mass would be from the bond energy! Same would go if you somehow managed to bind photons together.