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:
Enemies with missile/bomb or drone weapons have a limited supply of these, so if for some reason you're forced into a long fight, and you don't want to jump away from the battle, you only have to deal with limited number of shots from these weapons/limited ammount of drones used (this is generally more relevant in case of on board drones like anti personel, boarding, and repair drones)
However, waiting for them to run out of missiles may end up being very costly.