Yes, it is possible
- But it is not what sane solution should be
How big would they need to be?
- Depends on how much energy you wish to have, but bigger than solar panels.
What speed would they spin at?
- Limit are defined by strength of material you have build it from and size of construction. To be efficient, fast. IDK more.
How much power would they generate?
Depends on how big they are, but less than solar cells and other solutions.
Also depends on mass of construction, how close to star they are, how deep they are in gravity well(deeper they are, better and more efficient per kg they work)
Would they be blown out of their orbits?
Whole point of such system is based on the fact that they do not have natural orbit, and they hover using force of solar wind near star, which counteracts attraction from star, which(attraction) allows to actually generate energy.
They are called Statite, I'm not very familiar with them, but you may probably even find something similar to your idea. It is perfectly valid idea, it is just not mass efficient and have lots of other problems, and only positive thing I may say about it, is that it is perfectly capable to work.
That fact that solar wind blows in one direction(from star), does not mean Solar Sail Wind Turbines should rotate in one direction, and direction can be regulated by angle of blades. And system of 2 such Propellers is perfectly capable to keep momentum. Theoretically you even do not have to have 2 propellers for this system to work, just change rotation periodically, and use massive core (I fail to describe it in English, so just use 2 Propellers it is simpler to describe).
As note2, efficiency
Efficiency depend on speed of rotating blades. Faster this thing rotates, more energy it will generate, more efficient it will be. Work is equal force multiplied by distance, Power is equal force multiplied by velocity.
If we take same sail 800x800m as in Michael Karnerfors answer and same force 5N, then to get 900MW with this "blade" it have to have velocity 180'000'000m/s it is almost 2/3 of speed of light. Centrifugal force in this situation will be a problem, would be construction bigger(light years big), this would be less a problem. But "blade" not necessary have to be thin foil, it can be a plasma, and with magnetic field this plasma can be contained at this velocity, in similar size construction(km sizes or less). Although with small size(km's) and not dense plasma we will have other issues(plasma will be to transparent, so we should have ticker layer of that plasma, or something like that)
It still can have exterior of big Solar Sail Fan, where blades used to change angle of light on which it hits the plasma rotor. Faster it(plasma fan) will rotate, bigger red shift of reflected light will be.
But eventually it will be just particle accelerator, which uses light to accelerate particles(plasma) inside it.
So as simple construction SolarSailTurbine may be not the most efficient solution, but more sophisticated design may improve situation.