Intrepid Rabbit

The token created by the offspring ability isn’t “cast”, so abilities that trigger when a creature spell is cast won’t trigger for the copy.

The token copies exactly what was printed on the original creature and nothing else, except it’s a 1/1 (unless that creature is copying something else; see below). It doesn’t copy whether that creature is tapped or untapped, whether it has any counters on it or Auras and Equipment attached to it, or any non-copy effects that have changed its types, color, or so on.

Many creatures with offspring abilities have other abilities that refer to them as “this creature” rather than referring to them by name. This difference is for clarity purposes and does not change the function of any of these abilities.

If the spell resolves but the creature with offspring leaves the battlefield before the offspring ability resolves, you’ll still create a token copy of it.

In the rare case where the original creature is copying something else when the offspring ability resolves, the token enters as whatever that creature copied, except it’s a 1/1.