1. Do the PCs know the initiative of the adversaries before rolling theirs?
I don't recall if the rules explain that explicitly, but I usually get everyone's initiative rolls including the NPCs at the same time then figured out the order from there. I think the only difference it would make is knowing how to spend their free maneuvers generated by Advantages/Triumphs. So I suppose the answer is however you wish to play it.
2. When dealing damage when do the PCs know if they killed/knocked out the adversary. Before using their advantages or after?
Again, this isn't stated in the rules as far as I know, but I usually calculate and apply damage first and before I have my PCs spend their advantages and Triumphs. It doesn't break our game knowing if they should activate a critical injury or use that advantage to recover strain because the adversary just died. In conclusion, again it's up to you.
3. The PCs successfully stole Krayt Fang, but Trex is still alive. It would be very interesting to know how they can register the ship for themselves?
There's an organization in the EU known as the Bureau of Ships and Services (BoSS). It's been explained that BoSS keeps records of transponder codes and ship names as well as the individuals registered to that ship.
Finding a slicer to slice into the BoSS records to change the registration names or
blackmailing convincing a BoSS employee to do it for you would make for a fun session.
A good rule of thumb would be, if it's not explained in the movies, make it up. If you don't want to make it up. Try looking around in Wookieepedia.