You're absolutely right that we could choose all sorts of other wave shapes for many types of waves. Ones that travel at fixed speeds, like light in a vacuum, can be written as combinations of sine waves, or of square waves, or of smooth pulses, or of little wavelets, etc. All those descriptions are equally valid, despite what some teachers say.
Things change when the speed is different for different waves, like for light in glass. Then only some special shapes of wave will propagate without changing shape. It's far simpler to use these shapes to describe the propagating light. Those special shapes, the ones that propagate without changing shape, are the sine waves. Since most waves in most media have some of this dependence of velocity on shape, we get used to using the sine waves as the basis for our description of almost all waves, even in the cases where it's not really needed.
I'm not sure what you mean by "the path of a particle.: Does this refer to what path say a water molecule takes as a wave passes it? Or are you trying to picture the wave as a set of propagating particles?
For you first question, were you looking for a more general description or just the discussion of sine waves?
Mike W.
(published on 02/12/2013)