Why did Kyle have to shoot Aya to keep his cover for those last couple of minutes? He already had all the information he needed.
Watching the scene again, it's because No. 9 was right behind him and was watching the whole scenario. If he tried to suddenly turn around at that moment and shoot No. 9, there's no guarantee he wouldn't have been run through with 9's machete. By shooting Aya (and it was more more of a grazing shot, actually), 9 dropped his guard long enough to take a step back and turn toward Eve instead (important because now 9 is no longer looking at Kyle). This gave Kyle a crucial extra second to turn around and pop him in the head.
Why would Aya stop to have a shower and a nap with only a flimsy motel door between her and the ANMCs running all over town?
The shower is understandable, she's had a long and rough day, not to mention that this is the same day where she had to drive from Los Angeles to Nevada (easily a 5 to 6 hour drive). As for the nap... earlier in the game, after defeating No. 9 in the abandoned house, she loses consciousness after her Heroic B.S.O.D.. But she happens to wake up right when Kyle shows up with a gun in hand. Her mitochondrial abilities may give her Super Senses that allow her to sense when she's in danger while asleep or unconscious and to Wake-Up Fighting.