Here in New Zealand, we have many native species of birds, insects, frogs, lizards and the like that thrived when our islands were cut off from the rest of the planet, but that have become extinct, or are in imminent danger of being so due to introduced predators such as rats, stoats, hedgehogs, ferrets, cats etc. etc.
It leads to the bizarre situation that conservation here is largely about killing things.
Makes for scary reading!
We use a bigger version to catch possums. Its hard to tell how many we catch though, as the feral pigs like to carry off and eat the dead ones. Its really a fascinating logistical problem trying to eradicate a broad spectrum of pests.
That being said, wild rats are very destructive because of their hardiness and intelligence. Raccoons may be the top animal for intelligence in urban environments, but they can't match rats for their destructive potential.
Many predators live in balance with the population of their preys. When the predators are too successful, the prey population will diminish, which causes the predator population to shrink, which is good for the preys, etc.
But omnivores like rats can wipe out an entire prey population and then just switch to some other source of food.
Weta are a bit like large, nocturnal crickets. There are more than one variety, but you're probably thinking of the giant variety.
We get a relative of them here in the US, usually called cave crickets or camel crickets (here in VA).
Also we've stopped those pesky Victorians from introducing new species. Its a good thing they never introduced wildcats to control the stoats, weasels and ferrets - which they introduced to control the rabbits - which they introduced to give them something to shoot at on quiet Sunday afternnons.