 What is this photophoresis force? Why a new big word?

In the Epstein regime (aka free molecular flow) photophoresis is just what you may otherwise think of as Epstein drag, but now taking into account the surface temperature of the dust particle and the "accommodation" interaction between the surface and the gas molecules interacting with it. When there is a variation in the dust particle's surface temperature, the drag force is not just proportional and opposite to the particle's motion, there is a component related to the direction of the temperature gradient on the particle which is independent of the particle velocity (as long as that velocity is small compared to the gas particle thermal velocities of course). This also means photophoresis is closely physically related to the Magnus effect (in the free molecular flow, or low Knudsen number regime) and the drag torque on a spinning particle.
There are many books aimed mainly at aerospace with the title "Rarefied Gas Dynamics" which introduce gas moleculesurface interactions (link to example by C. Shen 2005). Those are good places to look for an introduction to energy and momentum accommodation coefficients, which play a crucial role in determining the forces exerted on a surface by gas molecules. I also recommend a little nugget Free Molecular Flow over a Rotating Sphere (C.T. Wang, AIAA, 1972, 10 (5)) which handles Epstein drag and lift (Magnus effect) and illustrates the importance of the accommodation coefficients. Here you'll also find that the lift on a spinning sphere in the free molecular flow (Epstein) regime is the opposite direction from the lift in the fluid (including Stokes) regime, which might come in handy if you are playing baseball or soccer in space.
 How does a large scale temperature gradient result in a force on a tiny particle?

Here's a handwaving way to think about it, aside form the equations. The problem contains three scales. They are well separated. The largest is the mean free path of thermal radiation photons in the optically thick medium. The middle one is the mean free path of gas molecules. The smallest is the radius of the particle that the photophoresis acts on. The particle "sees" by photons the temperature difference over the scale of a photon mean free path (an optical depth). The particle "feels" by colliding gas molecules the local gas temperature. As the particle is smaller than a gas molecule mean free path, the gas temperature is regarded as constant over that scale. So, even though the particle in sitting in a what is locally a constant temperature gas, it can be deferentially heated by a much larger scale thermal gradient.
 What about particle spin?

The basic answer is that in the Epstein drag regime (free molecular flow) spin is damped on the timescale of the stopping time for linear motion relative to the gas. This is treated in Epstein's original paper. To put it in brief, in Epstein drag, the force is exerted by the reflection (or adsorption/desorption) of gas molecules from the particle surface. These reflections are not all pure specular reflection (like form a mirror)  many are diffuse, in that the direction is scattered from the incoming direction. Diffuse reflections exchange momentum in a direction tangential to the particle surface with the gas. Thus, over many gas molecule interactions, a spinning particle exchanges angular momentum with the gas. These particle reflections are the same ones which case the linear drag, so the rotational and linear stopping times are related. Linear drag experiments yield measurements of the mix of diffuse and specular reflections  for example Blum et al. (1996) find 76%/24% diffuse/specular reflections.
A more advanced answer is treated a bit in the appendix of Paper II, and you can also convert the discussion into terms of the momentum accommodation coefficients. See the links in the top point on this page for more on accommodation coefficients.