Is it possible to wait for an enemy ship to run out off missiles and/or drones so that I can focus killing their crew with less pressure? If so, are the quantities fixed or random within some range?
Yes, they can run out of missiles as well as drones. It seems to be somewhere between 5-15 (random I assume) missiles/drones.
You can tell when they run out of missiles because their missile weapons will be powered up (you can see their power usage with Rank 3 Sensors) and charged (the graphic on their ship will have green lights) but they will never be able to fire.
The rebel Flagship (the boss) however, do not require missiles to fire its weapon as it is an augmentation, similar to the artillery beam.
Source: the wikia
According to the wiki, they are limited:
However, waiting for them to run out of missiles may end up being very costly.