Let's say the phrase I want to write is "all of the Mike Tysons, Donald Trumps, and Morpheus's in the world." I'm guessing that the first two names do not need one and the third does. Is this correct?
One never forms plurals out of English words via an apostrophe. The apostrophe indicates a possessive, not a plural.
The English plural of Morpheus is simply the regular plural Morpheuses.
What you are using is one form of posessive, like "Mike's shoes", or "Morpheus' eyeballs".
Instead you need a plural. Some words don't have a plural.
Some have an awkward plural, that comes from the originating language, like "Ox -> Oxen".
The plural of Morpheus is Morpheuses, I'd say. And yes, it's an awkward plural.