Do free protons and neutrons absorb much radiation? To affect astronomers' observations? If so, at what wavelength(s)?

You cannot have free protons without electrons. Plasmas, in general, are electrically neutral.

It is usually electrons that dominate the scattering (note that a point-like charge cannot absorb a photon and conserve energy and momentum) in a plasma at low photon energies. That is basically due to their much lower masses (classically you can think of the electric field of the incoming light accelerating the particles). NB. Electrons in the electric field of ions can emit or absorb photons in a process called (inverse) bremsstrahlung.

High energy (>MeV) photons are capable of causing transitions in the energy levels of nuclei, so can be absorbed. At even higher energies the photons can interact with a nucleus and create particle/anti-particle pairs - pair production.

Very high energy (>100 MeV) photons can scatter from protons and neutrons because these particles have internal quark structure. It is even possible for photons to be absorbed, creating short-lived 3 quark resonances.