Would it be acceptable for me to use a large universe for different stories? One story involving certain characters from my fictional universe, and then another story using different characters, and the events occur in a different area of the same universe?
So long as you have a rich and interesting universe, write as many stories as you can think of within it.
The great thing about worldbuilding is that you open up so many opportunities for story. Most of it will be backstory to help create a real universe for your first book. But if that backstory is rich enough, then it should be full of stories itself. There's so much fun you can have with a fully-functioning, fictional universe.
I'm still writing the first novel set in my fictional universe, but I've got about eight years worth of worldbuilding behind it. Because of that, I've currently got about twelve other novel ideas planned, all set in the same universe, spread out over ten thousand years.
Nothing beats getting deep and dirty in a well-made fictional universe.