Welcome to the Forums rtgwork!!
As for banish, the rules say that teleporting,push/pull, is entering in a zone. So I would say yes.
Pg.24 Traps side bar-An enemy creature might enter the zone by taking
a move action, or it might get Pushed or Teleported
As for summoned, this is being worked out right now, and we do not have an official answer. Im sorry for the inconvenience. My group plays with the assumption that summoning is in fact entering a zone. It would be the ONLY thing in the game to not be considered entering, and in other games we play, when something is summoned it is entering the square or zone or board, what have you.
This is from the FAQ:
The Mangler Caltrops only attacks creatures that enter into the zone. This is by any means -Pushing, teleporting, or just walking right in.
It does not attack creatures who are in the zone when it is cast, or who remain in the zone. It does not attack creatures when they exit the zone. If they leave the zone and come back in, it will attack them when they enter the zone again.