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Photons have momentum. If you're collecting light that's hitting your craft always at the same angle this momentum has to be transferred somewhere, and your craft is this somewhere as there is nowhere else for it to go.
Yes they do. Wikipedia points out that the force is (1/c) times the power, and helpfully converts 1/c to 3.34 Newtons per Gigawatt. The article also helpfully does the calculation for the solar radiation near 1 AU (i.e. Earth) and comes to a value of ten Micronewtons per square meter.
If one wants to use this force, the best thing one can do is have a very large and very light mirror, which is better than first taking the momentum of the suns photons on your solar collectors and then sending a small fraction of that momentum out in the direction you actually want to accelerate in. This is not completely hopeless: metallized Mylar foil might weight some 50 micrograms per square meter, so a space craft where most of the mass is in the foil might accelerate at 40 centimeters per second squared (though there are some constraints on the direction, similarly to sailing). Of course, having a spacecraft with two hectares of foil per kilogram of payload might be difficult from an engineering point, and micrometeorites might become a problem. I would probably play a Kerbal mod which adds Kerbol radiation pressure and giant sails, though.
Or you could actively shine an Earth- or Moonbound laser on your spacecraft.
In general, there is a tradeoff between getting the most momentum out of your propellant mass, which benefits from higher exhaust velocities on the one side (with photons being the optimal choice, and ultrarelativistic ions only slightly worse) and getting the most momentum per energy invested, which favors throwing out a huge mass at minimal velocity. For propulsions where the energy source is decoupled from the reaction mass, such as ion drives, the sweet spot seems to be at a mere 20-50km/s -- which is far away from the 300,000km/s you would get with photons.
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