An example that shows how Romans gave away citizenship rights in measured amounts can be seen when we look at the difference between the Latini and the Foederati.
The Latini were people who were born outside of Rome but on the Italian peninsula. Their citizenship allowed them to do business with Rome, and travel and live within the Empire. However, they were not allowed to have a Roman marriage. The Foederati, on the other hand, were the citizens of states that had obligations towards Rome. They were given limited citizenship rights in exchange for military service.
No, it is false that Imperialism was the name given to the idea that Americans should expand across the nation to the Pacific Coast, since what you're referring to was known as "Manifest Destiny".