Rulings

Any enters abilities of the copied permanent will trigger when the token enters. Any "as [this permanent] enters" or "[this permanent] enters with" abilities of the copied permanent will also work.
The token created by Applied Geometry copies exactly what was printed on the original permanent and nothing else, with the listed exception (unless that permanent is copying something else or is a token; see below). It doesn't copy whether that permanent 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 power, toughness, types, color, or so on. If it is a Vehicle, it is not crewed. If it is an Equipment, it is not attached to any creature.
If the copied permanent is a token, the token that's created copies the original characteristics of that token as stated by the effect that created the token, with the listed exception.
If the copied permanent is copying something else, then the token enters as whatever that permanent copied, with the listed exception.