Is scattering efficiency of dust related to albedo? I know that the scattering cross-sections of dust in comets depend on the scattering efficiency. A higher scattering efficiency leads to an increase in brightness. Therefore, does it mean that this will also increase the albedo or these quantities are not related at all?
There will be a relationship but not a simple one. If we model a uniform slab of material by a complex dielectric constant $n + ik$ then we can calculate three things; reflection, absorption and transmission. If we model a dust particle as a uniform sphere of the same, then the far field electric amplitude scattered from an incident plane wave on it will depend on the complex dielectric constant. The reason we can't just use reflection, absorption and transmission is that those are the amplitudes squared, and so phase information is lost, so you can't easily work your way backwards.
There is also the angular distribution to worry about, which will be different for each size and also for each complex index of refraction. So two identically shaped and sized dust particles; one from a high bulk albedo material and one from a low bulk albedo material could end up with a variety of different relative scattering efficiencies.
A cloud of either one might attenuate light from a star, depending on the details, and one might scatter light more strongly at some specific angle than the other.
There might be some rules of thumbs for certain kinds of dust particles made of materials most likely to be observed and discussed by astronomers, but I don't think there are any first-principles-based generalizations that always apply.