The ability is granted as part of the copy effect, which means it will be removed by any effects that would remove abilities regardless of when the effect was created.

If an Unstable Shapeshifter is on the battlefield when another Unstable Shapeshifter enters, the first one gets the second one's triggered ability, so it will copy the next creature that enters twice. It will only do so the next time a creature enters, because the second "becomes a copy" effect to resolve will erase the extra triggered ability.

The copy effect is not targeted.

If a creature enters that has some characteristic or ability set when it enters, the copy does not get to make that choice (because it is not entering. All values are zero and all checks for choices consider the choice not to have been made.

If the creature has power/toughness that is continuously recalculated, so does this copy of it.

When it changes "shape" to a new creature, any "enters" abilities of the creature do not trigger.