If my opponent uses Pokemon Catcher or Escape Rope while Dust Island is in play, is my new Active Pokemon still poisoned.
No, it is not. Dust Island only works when a Trainer card is used to switch your own Pokemon; it does not apply when the Trainer card is played by your opponent.
Source: Unbroken Bonds FAQ (2019-04-19), TPCi Rules Team (2019-05-02)

Relevant cards

Note: All applicable cards may not be listed below.
SM10_168
SM10_168