Spanwise flow at the surface of the wing is not required for induced drag to occur. If you can make a compelling case to the contrary, I would like to hear it.
Is has to start somewhere, and that somewhere is at or near the leading edge of the wing. It just happens to be most "visible" after it has left the wing surface (the pretty picture I posted). As the aircraft goes faster, it "leaves behind" this spanwise flow and becomes more efficient... as is evidenced by reduced induced drag at higher speeds.
Obviously, if a tip vortex exists, there will be spanwise flow somewhere far from the wing. Yes, reducing the tip vortex strength will reduce induced drag. The only problem with that is that you can't reduce the tip vortex strength without reducing lift.
Not really... induced drag is basically an inefficiency of a finite span wing compared to an infinite span wing. So, by having a tip vortex, you are not going to get increased lift/drag by having a more powerful vortex. Conversely, you will not lose lift/drag by reducing the strength of the tip vortex. This strength though, must be viewed in terms of the overall lift that the wing is producing at the same time. Increasing the AR is one way of doing this, so is modifying the tip geometry.
Methods that reduce induced drag without reducing lift (significantly) do so by modifying the effect of the tip vortex on the wing, usually by shifting the tip vortices out, or increasing the vortex span. This is why it pains me to hear people attribute induced drag to the tip vortex, rather than the downwash. It leads to this fallacy that tip vortices can be somehow impeded or reduced, which can really only happen by reducing lift.
The thing that I don't think you understand is that the downwash itself is a
result of the tip vortex, and the tip vortex is a result of spanwise flow. Remember, spanwise flow is any component of the flow coming off the wing, in the span direction. Downwash is a result of the inefficiency of the wing and is not necessary for lift.
You mentioned before that
induced drag is a short name for
lift induced drag. However, some sources I've recently consulted (as a result of this discussion) also call it "vortex drag".