The copy effect created by the last activated ability doesn't have a duration. It will last until Thespian's Stage leaves the battlefield or another copy effect overwrites it. The permanent will no longer have the first ability of Thespian's Stage.

A land's copiable values are those printed on it, as modified by other copy effects. Counters and other effects aren't copied. Notably, if you copy a land that is also a creature because of a temporary effect (such as Celestial Colonnade), Thespian's Stage will become just the “unanimated” land.

No enters-the-battlefield abilities of the land Thespian's Stage is copying will trigger. Thespian's Stage was already on the battlefield.