They could use laser based comms from one satellite to another:

https://en.wikipedia.org/wiki/Laser_communication_in_space

But optical signal amplification is a trick that really only works when you are on a very long run where the signal gets attenuated to the point that you need to boost it. There is no routing option in such a system, it is essentially point-to-point with in-line amplifiers.

So even if they may use laser based comms I don't think optical amplification will be used, the more likely avenue is to decode the optical signal, decide where it has to go next and then fire off another burst.

Alternatively they may use radio waves instead of lasers, I'm not sure what system SpaceX will use for this particular system, given the accuracy required for laser based communications I would assume they will use radio.

Are you talking satellite-to-satellite relaying? Assuming they're all moving relative to each other, then probably not, unless the laser transmitter or receiver are moving as well. And moving parts wear down and require power, so you'd want to avoid them on a satellite if possible. However, if parts of the constellation are following each other in exactly the same orbital path, then those parts might be able to optically relay without moving parts.