I know that nothing can exceed the speed of light c. So is it consistent with special relativity because it is not a physical object that is moving at 1.60c? I'm asking because I'm really not sure if I'm correct.

And note that for v<<c, you get back the old "velocity addition" rule that we are familiar with. It shows that the relativistic velocity addition is the more general rule.

Are you asking how the space can be increasing by that much & yet none of those two spaceships measures a superluminal speed ... ? Ordinary velocity addition doesn't apply.

Well, the simple answer here is that "separation rate" is not a speed. Speed is measured between two objects, not three. So it is perfectly fine to say that according to the 3rd observer, the two spacecraft are separating at 1.6C. That isn't contradicting SR because it isn't saying anything about a speed.

1. SR precludes one massive object from moving at a speed equal or larger than c.(photons having zero mass move at c)

2. In your example you are dealing with two separate objects closing on each other at a net speed v>c. SR does not preclude this, actually speeds as large as 2c are acceptable in SR.

This example above simply says that if a rocket flies in the +x direction at 0.8c and you are observing it from another rocket flying at 0.8c in the -x direction, you would measure the other rocket receding from you at 0.976c.
The observer in your original problem sees you and the other rocket receding away from each other at 1.6c.
The difference is quite clear now, right?

Yes, so if I was in one of the rockets, I would see the other moving traveling away away from me at 0.976c.

The external observer on the ground would see the two rockets traveling away from each other at 1.6c. This is possible because special relativity permits two separate objects to have a net speed greater than c, but not one object.