You may have Progenitor Mimic enter the battlefield as a copy of any creature on the battlefield, except it has "At the beginning of your upkeep, if this creature isn't a token, create a token that's a copy of this creature."
Progenitor Mimic copies exactly what was printed on the original creature and nothing more (unless that creature is copying something else or is a token). It doesn’t copy whether that creature is tapped or untapped, whether it has any counters on it or Auras attached to it, or any non-copy effects that have changed its power, toughness, types, color, or so on.
If you choose to have Progenitor Mimic enter the battlefield as a copy of a creature, the triggered ability it gains will become part of its copiable values. For example, suppose Progenitor Mimic enters the battlefield as a copy of Runeclaw Bear, a 2/2 green Bear creature with mana cost . The resulting object is a 2/2 green Bear creature named Runeclaw Bear with mana cost and with “At the beginning of your upkeep, if this creature isn’t a token, put a token onto the battlefield that’s a copy of this creature.” If another Progenitor Mimic enters the battlefield as a copy of that creature, it will be a Runeclaw Bear with two instances of the triggered ability.
If Progenitor Mimic somehow enters the battlefield at the same time as another creature, it can’t become a copy of that creature. You may choose only a creature that’s already on the battlefield.
Any enters-the-battlefield abilities of the copied creature will trigger when Progenitor Mimic enters the battlefield. Any “as [this creature] enters the battlefield” or “[this creature] enters the battlefield with” abilities of the chosen creature will also work.