It is likely the Bullhead was on standby in the area for the purpose of rapid evac. (I don't think she concealed that, let alone the noise it makes.) Roman or Neo could've signalled it for evac or it could've been tracking them. (Considering Roman left the warehouse with the Paladin, it's probable that Neo took the airship to follow him.) As for how they managed to board it in that short duration… I'm not entirely sure of that myself. Either the illusions are capable of relaying or play back audio, or they moved very quickly. (Remember that Blake and Sun managed to outpace the Paladin, which also kept pace with highway traffic.) Also, powered winches are a possible factor in their boarding of the Bullhead.
As for why Emerald was needed, remember that different situations require different tools. Let us use the Vytal Tournament for example.
Considering that Cinder needed to sow fear and panic to attract the Grimm, having Neo project an illusion on the size and scale of an arena may be too much for her to handle and it is possible other forms of observation could render the illusion pointless. (Motion tracking, thermal imaging, acoustic detection, Ozpin's magic cameras that can see and follow people, etc.) It is also possible that she could have been out of range to project it on subjects in the arena. (And for that matter, assuming that it affects a radius around her, it would likely encompass other witnesses and possibly not affect them as they are inside this hypothetical radius and render the point of the illusion moot.)
Emerald's use of her semblance on Yang so that she "crippled" Mercury, however, not only accomplishes the task of convincing the public of the ruse, but also convinces the fall guy (in this case, Yang) that she perpetrated the act and thus demoralizes her. (Unstable emotional and psychological states can adversely affect combat performance.) Plus, considering Mercury needed to "disappear" from the tournament, a fragile external illusion would make it difficult for that to happen. (Not sure how difficult it is to simulate damage from weapon's fire with that semblance.)
Neo's illusions would also not work in the Penny vs Pyrrha fight, as her illusions would be destroyed. (Assuming Penny being diced by her own wires was the plan and the scenario played out as such, regardless.) Again, Emerald causing Pyrrha to hallucinate Penny's attack (and eventually accidentally kill her) not only accomplished the goal, but again, also demoralized the intended scapegoat.
Finally, in concerns to that last fight, Neo was most likely busy working on springing Roman from the brig.