Is there a difference between generating multiple numbers using a single random number generator (RNG) versus generating one number per generator and discarding it? Do both
I would recommend using a single generator multiple times. As far as I know, all the generators have a state. When you seed a generator, you set its state to something based on the seed. If you keep spawning new ones, it's likely that the seeds you pick will not be as random as the numbers generated by using just one generator.
This is especially true with most generators I've used, which use the current time in milliseconds as a seed.