Riker and his team transport directly aboard the Hathaway, which is supposedly without power. There could be breathable air, but a ship abandoned without life support would be too cold for them to board without freezing. Remote access can certainly take place and obviously there is power or there would have been no lights when they beamed over. Obviously they would have activated life support ahead of time.
ST:TNG Technical Manual says that antimatter cannot be beamed through a transporter without making extensive modifications to the circuitry. Yet Wesley took no such precautions, and was still able to beam it safely. On the other hand, Wesley is a total genius and his project could have involved the creation of a means for containing antimatter that can be beamed. For the story to work, that would be exactly what happened.
Riker's boarding team are carrying First Alert flashlights.
Several times during this episode, some of the computer displays along the back of the Enterprise bridge are covered by what appears to be matte black material, apparently to mask reflections from studio lights. They're especially evident at the start of the battle simulation. As the Ferengi attack begins, they mostly vanish, only to return as Picard communicates with the Ferengi.
Capt. Picard addresses Ensign Burke as "Lieutenant Burke" at one point.