Is deep-space radiation greater in the Sun, or in the shadow of Earth?

Where is the deep-space radiation higher - on the Sunward side of Earth, or on the side in Earth's shadow? I guess this may also be rephrased to read - what fraction of shielding from deep cosmic rays within the Solar System is provided by the Solar wind?

I think you are confusing a few things.

Radiation is a generic term that applies to lots of different things. For instance, all forms of electromagnetic waves are a type of radiation, but not all electromagnetic waves are harmful (most harmful radiation is called ionizing radiation... I say most because microwaves can still kill you if intense enough but they are not a form of ionizing radiation). Free particles moving in space are also a form of radiation. Thus, all the particles comprising the ionized gas, called a plasma, that is the solar wind is considered radiation.

Thus, the solar wind itself does not shield us from anything. The interaction of the solar wind and the incident interstellar medium produces a termination shock, heliosheath, and heliopause at the edge of the heliosphere. These boundaries can have large gradients in the magnetic field, which can deflect charged particles (e.g., most types of cosmic rays are comprised of charged particles). However, none of these boundaries are even seen by most forms ionizing electromagnetic radiation (i.e., UV light and higher energies). Meaning, these boundaries are (mostly) transparent to energetic electromagnetic radiation.

To answer part of your question, you would be shielded from many forms of ionizing solar radiation if immediately behind the Earth, but not all. Some of the intermediate energy charged particles would actually bend around Earth because of Earth's magnetic field. However, the fractions of particles that can do this are infinitesimal compared to the fluxes found in the radiation belts. Thus, just because you are near Earth does not mean you are safe.

Is a cosmonaut at greater health risk on the Sunward Side, or in Earth's shadow? (This second part may fit better on Biology SE but since it is contingent on the former I post it here)

I think the biggest issue for space travel is not direct irradiation at LEO from the sun but the significant increase in solar radiation one would experience if outside Earth's magnetosphere.

While inside, say, the international space station (ISS) astronauts are protected from most forms of ionizing electromagnetic radiation. In fact, most areas in the ISS are pretty good at stopping protons below ~1-10 MeV and there are shielded areas used during solar storms (e.g., solar flares) that provide protection to higher energies. It is, however, impossible to stop all radiation so they compromise based upon flux levels, expected dose rates, and mass considerations.