I'm not a conservation expert, but think it's safe to say that the answer to your question is a bit complicated. We have a national goal to be free of a list of predators including those by 2050 - it seems achievable, but only with a lot of work and political willpower. Stoats for instance can be killed with traps that cats can't get in to, so that's a useful tool in the more accessible areas where we still have pet cats to contend with. One of the more immediate issues in the big picture, is a small but vocal faction that's violently opposed to using a poison that is key in these efforts. To give some flavour, check out this recent and very well-done series: https://www.rnz.co.nz/programmes/fight-for-the-wild
No, we don't. The only requirement is a chip + rabies shot if you are taking your cat, dog or ferret with you across other European countries (per directives 998/2003 resp. 576/2013).
> My ex lives there, has lost her cat, it was captured by animal control and she had to pay a 300€ fine at the animal shelter to get it back.
That is not a fine by the government, it is a charge from the shelter for the trap, medical checkout and accomodation cost.
What about stoat, weasels and ferrets? I imagine one of those kills at least three fold more wildlife and birds than a cat, not counting eggs. Mustelids basically drove the kakapo to extinction.