In my opinion it is not necessary to evoke the theory of relativity or quantum physics to explain how light can have momentum but not mass. In the 19th century, it was already known that light can collide with matter; a beam of light can set a small wheel (in vacuum) rotating. read more
"momentum is the product of mass and velocity, so, by this definition, massless photons cannot have momentum" This reasoning does not hold. Momentum is the product of energy and velocity. read more
Yes, if we define the source of gravitational effects to be mass, we can see that light has mass in that sense. However, if you tried to use that force equation you gave, you'd calculate some bending of light in a gravitational field, but it would only be half the observed amount. read more