Yes, if you chose to.
the wonderful thing about birds is how easy it is to control their reproduction. with mammals, breeding or mating pretty much means you have babies incoming. With chickens, they can mate all they like, but until
you decide to incubate eggs, there are no babies from all the mating.
So, if you currently have a flock of Reds, and introduce other breeds of hens, yep, your Red rooster will happily mate those hens. Their offspring, should you chose to incubate their eggs, would be mixed breed birds.
But, if you wanted only Red chicks, you could easily figure a way to separate a few Red hens and collect just their eggs and incubate those, giving you pure bred chicks.
Having a rooster of a different breed makes things a bit trickier. Hens are capable of storing sperm from a rooster a good 2 weeks. So, if you had say an Australorp rooster in with your Red hens, you'd have to find a way to separate them at least that long, while still letting them be exposed to the Red rooster to fertilize the eggs you wanted to hatch.