will a Sears-Haack body straighten itself out if "launched" sideways?
will an object moving at high speed find and stay in its most aerodynamic orientation?


No. It will most likely tumble around. With the center of gravity behind its center of pressure, a body of rotation is inherently unstable.


  if not, will a major control surface like fins make it do so?


Yes. That is the reason for those large surfaces at the rear end of airplanes. 

This answer is about directional stability over Mach. At supersonic speed directional stability drops with increasing Mach number.

This answer covers a similar topic; now it is about the proper location of fins at supersonic speed.