The Interplanetary Laser Jousting Tournament

It is said that two spaceships going away from each other at very high speeds are able to see light from the other travelling at the proper speed of light because space contracts and time slows down. What would happen if these two spacecrafts passed each other? Would space suddenly invert and time change speed? How would a spaceship travelling between two light sources, such as stars, see light from both going the right speed at the same time?
If, on the other hand, light travels at a broad range of speeds and we can detect only that which is going the right speed for us, then where is all the other light hiding?
An electron in a cyclotron that is accelerated toward a target at close to the speed of light gains mass. If the target were racing away at a near equal speed, would that extra mass turn out to not be there?
This book suggests an alternate theory to special relativity and a few experiments to prove it - or help us to learn something new.