Aprojectile leaves a gun at the same instant that the target is dropped from rest. if the projectile is initially aimed straight at the target, will it hit the target?

Depends on the mass of the projectile versus the object.

The force of gravity is dependent upon the mass of the object.  Therefore if the mass of the projectile and the mass of the object are exactly the same than both objects will fall at the same rate and will collide.  If the mass of the object is significantly larger than it will fall faster and the projectile will pass above it.  The opposite is true if the mass of the object is less than the mass of the projectile, in which case the projectile will pass underneath.

Lastly, this analysis excludes the force of wind resistance on the projectile and the object.  Under a complex model how the projectile travels through the air will have a separate impact that will alter its decent or rise outside of gravity.  For example, if the projectile is spinning it could "cut" into the wind and cause accelerated drop.  Think of a curve ball in baseball or a rising fastball.